1 CHAPTER 1 SYSTEMS OF LINEAR EQUATIONS AND MATRICES 1.1 Introduction to Matrices A matrix is a rectangular array that has the following form: 31 32 33 21 22 23 11 12 13 a a a a a a a a a - This is an example of a 3x3 matrix. A matrix with m rows and n columns is said to be a mxn matrix. If m=n then the matrix is a square matrix of order n. Example 1 2 2 −1 3 6 3 3 2 1 1 − 6 3 2 1 0 4 4 1 2 3 A matrix with only one row or one column is called a vector. 1.2 Matrix Operations • Addition/Subtraction Example 2 Let = = = c g d h a e b f .Thus, A B g h e f , B c d a b A Subtraction and addition of matrices are only defined for matrices of the same type.
2 • Scalar Multiplication Let .k is called a . kc kd ka kb , then kA c d a b A scalar multiple = = • The zero matrix is one in which all entries are zero. • Matrix Multiplication + + + + = ce dg cf dh ae bg af bh g h e f c d a b Example 3 Perform the following matrix multiplications (if defined). a. − − − 1 3 2 1 4 3 1 3 2 − − − 1 3 0 3 2 2 1 1 0 b. (− 2 1 3) − − − 3 7 1 1 4 2 c. − − − 1 2 3 2 1 4 1 6 1 4 d. 0 0 1 0 1 0 1 0 0 − − − 12 7 39 23 9 6 2 4 14 If A is a mxn matrix and B a nxp matrix then matrix AB is a mxp matrix i.e.the product of 2 matrices is only defined if the number of columns of the first matrix equals the number of rows of the second matrix.
3 Example 4 = = 0 0 2 3 B 0 0 0 1 A Find: a. AB b. BA Notice that: AB BA Note: 1. Division of matrices is undefined. 2. Division by a scalar is similar to scalar multiplication,i.e. (A) k 1 k A = Exercise 1 If A A 3 = , express the following in terms of A : a. 6 A b. 20 A • Suppose A is a matrix. The transpose of A, written as T A is given by: where the rows o f A become the columns o f A and vice versa. T = c f i b e h a d g g h i d e f a b c T Note that: 1. ( ) T T T A B = A B 2. (A ) A T T = 3. ( ) T T T AB = B A 4. (kA) k(A )- k is a scalar multiple T T = Suppose A is a square matrix. 1. A is symmetric if A A T = 2. A is skew-symmetric if A A T = −
4 Example 5 − − = − = 2 3 0 1 0 3 0 1 2 ,B 5 6 3 4 2 6 1 4 5 A . Find T A and T B . • a. A square matrix in which all entries not on the main diagonal are zeroes is called a diagonal matrix. Suppose = dn d d 0 0 0 0 0 0 0 0 0 D 2 1 . Then 1 2 0 0 0 0 0 0 D 0 0 0 k k k k n d d d = b. A diagonal matrix in which all main diagonal entries are 1 is called an identity matrix. The symbol for the identity matrix is n I . Then the identity matrix AI = IA = A. c. A diagonal matrix in which all diagonal entries are equal is called a scalar matrix. d. An upper (lower) triangular matrix is a square matrix in which all entries below (above) the main diagonal are zeroes. Example 6 is upper triangular and 0 0 0 4 0 0 3 2 0 4 0 7 2 1 3 6 − − is lower triangular 5 1 3 2 0 1 7 0 0 5 0 0 3 0 0 0 .
5 • a. A matrix is said to be in row-echelon form if it has the following properties: 1. All zero rows, if there are any, appear at the bottom of the matrix. 2. In two consecutive non-zero rows, the first non-zero entry in the lower row appears to the right of the first non-zero entry in the upper row. b. A matrix is said to be in reduced row-echelon form if it has the following properties: 1. All zero rows, if there are any, appear at the bottom of the matrix. 2. The first non-zero entry of a non-zero row is a 1. This entry is called a leading 1. 3. For each non-zero row, the leading 1 appears to the right and below any leading 1’s in the preceding rows. 4. If a column contains a leading 1, then all other entries in that column are zero. The following matrices are in reduced row echelon form. 1 0 0 4 0 1 0 7 0 0 1 1 − , 1 0 0 0 1 0 0 0 1 , 0 1 2 0 1 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 − , 0 0 0 0 . The following matrices are in row echelon form. 1 4 3 7 0 1 6 2 0 0 1 5 − , 1 1 0 0 1 0 000 , 0 1 2 6 0 0 0 1 1 0 0 0 0 0 1 − . Note: 1. A matrix in RREF is necessity in REF, but not conversely. 2. RREF must have 0 before and after leading 1 while REF after leading 1 must 0 but before not.
6 Example 7 State whether each of the following matrices is in row-echelon form, reduced row-echelon form or neither. 0 0 0 0 0 0 0 0 3 2 2 3 , 0 0 0 0 0 0 0 0 1 0 0 0 4 5 0 0 1 2 3 0 , 0 0 0 0 0 0 1 0 0 0 0 1 , 0 0 0 0 0 0 0 1 0 0 3 4 0 0 1 2 , 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 2 0 0 1 0 1 0 0 0 , 0 0 1 0 1 0 1 0 0 , 3 0 0 0 3 5 0 0 5 2 1 0 , 1 0 0 0 1 2 0 0 1 2 3 0 , 0 0 0 1 2 0 0 1 2 3 0 1 2 3 4 1 2 3 4 5 . 1.3 Systems of Linear Equations A linear equation in n variables has the following form: a1 x1 + a2 x2 + ......+ an xn = b Example 8 The following equations are all linear. • x1 − 2x2 − 3x3 + x4 = 1 • x + 3y = 7 • y = 0.5x + 3z +1 • x1 + x2 + ......+ xn = 1 Example 9 Examples of non-linear equations are given below: • 3 7 2 x + y = • y = sin x • 3x + 2y − z + xz = 4 • x + 2y + z =1 • + = 0 x y e Notice that: 1. Linear equations do not involve any products or roots of variables. 2. All variables occur only to the first power and do not appear as arguments for trigonometric, logarithmic or exponential functions.
7 A system of linear equations can be written as: m m m n n m n n n n a x a x a x b a x a x a x b a x a x a x b + + + = + + + = + + + = ...... ...... ...... 1 1 2 2 21 1 22 2 2 2 11 1 12 2 1 1 Where n x , x ,......,x 1 2 are n variables; aij bj , are constants. The above system can be written in matrix form as follows: = m m m n n m n n b b b x x x a a a a a a a a a 2 1 2 1 1 2 21 22 2 11 12 1 m m m n n n a a a a a a a a a 1 2 21 22 2 11 12 1 is called the coefficient matrix of the system of linear equations. m m m n m n n a a a b a a a b a a a b 1 2 21 22 2 2 11 12 1 1 is called the augmented matrix of the system of linear equations. Example 10 Obtain the coefficient matrix and the augmented matrix for the following system of linear equations: 6 5 3 2 3 1 7 2 9 + − = − − = − + + = x y z x z x y z
8 A system of linear equations may have: a. Only one solution (unique solution) b. An infinite number of solutions or c. No solution A system that has at least one solution is said to be consistent. A system that has no solution is said to be inconsistent. In order to solve a system of linear equations, we perform several operations on the augmented matrix. These operations are called elementary row operations. The augmented matrix is thus changed to its row-echelon or reduced row-echelon form. There are 3 types of elementary row operations as follows: 1. Multiply a row with a constant. ( ) Ri → kRi 2. Interchange 2 rows. (R R i j ) 3. Add a multiple of one row to another row. ( ) Ri → Ri + kRj Example 11 Solve the following system of linear equations using elementary row operations. 3 6 5 0 2 4 3 1 2 9 + − = + − = + + = x y z x y z x y z Answer: x y z = = = 1, 2, 3 Example 12 Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form. Solve the system. 1. − 0 0 1 4 0 1 0 2 1 0 0 5 2. − 0 0 1 3 2 0 1 0 2 6 1 0 0 4 1 3. − 0 0 0 1 5 2 0 0 1 0 3 1 1 6 0 0 4 2 4. 0 0 0 1 0 1 2 0 1 0 0 0
9 1.4 Gauss – Jordan Elimination Gauss – Jordan(Gauss) elimination reduces a matrix to its reduced row-echelon form (row-echelon form) Step 1 Locate the leftmost non-zero column. Step 2 Interchange the top row with another row, if necessary, so that the top-most entry of the column found in Step 1 is non-zero. Step 3 If the entry that is now at the top of the column found in Step 1 is a, multiply the first row by 1/a in order to obtain a leading 1. Step 4 Add suitable multiples of the top row to the rows below so that all entries below the leading 1 become zeroes. Step 5 Now cover the top row in the matrix and begin again with Step 1 applied to the submatrix that remains. Step 6 Beginning with the last non-zero row and working upward, add suitable multiples of each row to the rows above to obtain zeroes above the leading 1’s. Example 13 Obtain the reduced row-echelon form of the following matrix using Gauss-Jordan elimination. − − − − − 2 4 5 6 5 1 2 4 10 6 12 28 0 0 2 0 7 12
10 Example 14 Solve the following systems of linear equations using Gauss-Jordan elimination and state the type of solution obtained. 1. 3 2 1 2 1 3 2 1 2 1 2 1 2 + = + = − = − x x x x x x 2. 2 2 2 1 2 3 0 1 2 3 1 2 3 1 2 3 − − + = + − = − + − = x x x x x x x x x 3. 2 6 8 4 18 6 5 10 15 5 2 6 5 2 4 3 1 3 2 2 0 1 2 4 5 6 3 4 6 1 2 3 4 5 6 1 2 3 5 + + + + = + + = + − − + − = − + − + = x x x x x x x x x x x x x x x x x x Example 15 1. Determine the value/s of k for which the following system of linear equation a. Has a unique solution b. Is inconsistent c. Has infinitely many solutions 3 2 2 3 3 1 + + = + + = + − = x ky z x y kz x y z 2. For what value/s of a would the system of linear equations below be consistent? 2 ( 2) 2 2 3 2 3 2 + − = + + = + + = y a z a y z x y z
11 Equivalent Matrices Two matrices are equivalent if one matrix can be obtained from the other by performing one or more elementary row operations. Symbol: A~B 1.5 Homogeneous Systems of Linear Equations A homogeneous system of linear equations has the following form: ...... 0 ...... 0 ...... 0 1 1 2 2 21 1 22 2 2 11 1 12 2 1 + + + = + + + = + + + = m m m n n n n n n a x a x a x a x a x a x a x a x a x Every homogeneous system of linear equations is consistent, since x1 = 0; x2 = 0;xn = 0 is a solution. This solution is called a trivial solution. If there are other solutions besides x1 = x2 == xn = 0 , they are called non-trivial solutions. Example 16 Solve the following homogeneous system of linear equations. 0 2 0 2 3 0 2 2 0 3 4 5 1 2 3 5 1 2 3 4 5 1 2 3 5 + + = + − − = − − + − + = + − + = x x x x x x x x x x x x x x x x Theorem Suppose a system of linear has m equations and n variables. If m<n, then the system of linear equations has an infinite number of solutions.
12 1.6 Matrix Inverse (I) Definition A square matrix, A, is said to be non-singular or invertible if there exists a matrix B such that AB=BA=I. B is called the inverse of A and we write B= -1 A . The inverse of A is unique. If = c d a b A , A is invertible if and only if ad - bc 0 and the inverse of A is given by: − − = c a d b ad - bc 1 A -1 Example 17 Find -1 B if = 1 3 2 4 B . Theorem If matrices A and B are both invertible, then, a. (AB) is also invertible. b. ( ) -1 -1 -1 AB = B A Proof (b) ( )( ) ( ) ( ) 1 -1 -1 -1 -1 -1 -1 -1 -1 AB B A I AA AIA AB B A A BB A = = = = = − Theorem If A is invertible, then a. -1 A is also invertible and (A ) A -1 -1 = . b. n A is also invertible and ( ) ( ) n -1 -1 n A = A . c. If k is a scalar,kA is invertible and ( ) -1 -1 A k 1 kA = . d. T A is also invertible and ( ) ( ) T -1 -1 T A = A .
13 Example18 If = 2 1 - 5 - 3 A , find ( ) ( ) T -1 T 1 1 A ,A , A and 3A − − . Suppose D is a diaganol matrix i.e. = dn d d 0 0 0 0 0 0 0 0 0 D 2 1 and di 0 for all I, then -1 D is given by: = dn d d 0 0 0 1/ 0 1/ 0 0 1/ 0 0 0 D 2 1 -1 1.7 Matrix Inverse (II) To find -1 A , reduce (A/I) to ( ) -1 I/A Example 19 Find the inverse of 1 0 8 2 5 3 1 2 3 and the inverse of − − 1 2 5 2 4 1 1 6 4 by row reduction.
14 Solving Systems of Linear Equations Suppose Ax = b . To find x , pre-multiply by -1 A : x b x b -1 -1 -1 A A A A = = Example 20 Solve the following system of linear equations by first finding the inverse of the coefficient matrix. 8 2 2 5 3 1 2 3 0 + = + + = − + + = x z x y z x y z 1.8 Elementary Matrices Definition An nxn matrix is called an elementary matrix if it can be obtained from the nxn identity matrix, n I by performing a single elementary row operation. Example 21 a. 1 R R 3R 2 E 0 1 1 3 I 1 1 2 = − ⎯⎯→⎯− ⎯→ b. 2 R 4R 2 E 0 4 1 0 I 2 2 = ⎯⎯→ ⎯→ c. 3 R R 2 E 1 0 0 1 I 1 2 = ⎯⎯⎯→ When a matrix A is multiplied on the left (i.e. pre-multiply) by an elementary matrix, E, then the product, EA is the matrix that results when the same row operation used to produce E is performed on A. Example 22 a. − − − − = − e f g h a e b f c g d h e f g h a b c d 3 3 3 3 0 1 1 3 b. = e f g h a b c d e f g h a b c d 0 4 4 4 4 4 1 0
15 c. = a b c d e f g h e f g h a b c d 1 0 0 1 Elementary row operations can be reversed. For example, 1 R1 4 1 R → is the reverse of R1 → 4R1 i.e.the 2 operations cancel out each other. Using the inverse operation on E will produce I. Using the inverse operation on I will produce -1 E . Question: What are the inverse operations for the following row operations? a) Ri → kRi b) Ri Rj c) Ri → Ri + kR j Suppose E is obtained from I by performing a single operation and E’ is obtained from I by performing the inverse operation. We find that: EE’=E’E=I i.e. E’= -1 E Example 23 The inverse operations from the earlier Example 21 are: a. b. c. Find the inverse matrices for E1 ,E2 and E3 using the above inverse operations. Every elementary matrix is invertible and the inverse is also an elementary matrix. Theorem Suppose A and B are row equivalent i.e. A~B and B is obtained from A by performing k elementary row operations. Then there exists k elementary matrices, 1 2 Ek E ,E , , such that: B = EkEk-1 E2E1A Example 24 B is obtained from A by performing 2 elementary row operations. Find elementary matrices E1 and E2 such that B = E2E1A and -1 E1 and -1 E2 such that A E E B 1 2 1 1 − − = .
16 = 0 4 3 2 6 8 1 3 7 A − = 0 0 6 0 4 3 1 3 7 B Example 25 Express − − = 3 8 1 2 A as a product of elementary matrices. Solution ⎯⎯⎯ ⎯→ ⎯⎯ ⎯→ ⎯⎯⎯ ⎯→ ⎯⎯ ⎯→ − − → − → →− → − 0 1 1 0 0 1 1 2 0 2 1 2 3 8 1 2 3 8 1 2 1 1 2 2 2 i 1 2 2 1 R R 2R R 2 1 R R R R R 3R From the above row operations, we obtain the following elementary matrices: − = 0 1 1 0 E1 − = 3 1 1 0 E2 = 2 1 0 1 0 E3 − = 0 1 1 2 E4 From the matrix product, E E E E A I 4 3 2 1 = , we solve for A to obtain: 1 4 1 3 1 2 1 A E1 E E E − − − − = This implies that A is a product of elementary matrices i.e. − − = − = − − = 3 8 1 2 0 1 1 2 0 2 1 0 3 1 1 0 0 1 1 0 3 8 1 2 A
17 Exercise 1 1. Compute AB and BA (if defined): a. , B (1 2) 4 3 A = = b. − − = − = 2 4 3 0 1 2 3 4 , B 4 0 1 2 A c. = − − − = 0 1 1 1 0 1 , B 7 4 2 5 3 0 A d. − − = − − = 3 1 1 6 , B 6 11 8 9 7 1 0 3 2 4 A 2. Find 8 A if . 0 1 1 1 A = 3. Consider the following matrix: − − = 0 2 1 2 0 1 1 1 1 A Show that A A 3 = . Hence, find A . 40 4. Find A ,A A,AA : T T T a. = − 7 5 0 4 6 0 A b. − = 1 3 2 A 5. State whether each of the following matrices is in row-echelon form, reduced rowechelon form or neither. 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 3 0 3 0 0 1 0 5 0 0 1 5 1 0 0 5 , 0 1 3 3 1 0 3 1 , − 0 1 3 3 1 9 3 3 , 0 0 0 0 0 0 0 0 0 1 1 0 3 3 0 1 3 0 3 0 0 0 0 0 0 0 0 0 0 , 1 0 0 1 , − 0 1 2 1 , 0 1 0 0 0 1 1 0 0 , 0 0 1 1 0 0 0 0 0 , 0 0 0 0 1 0 0 1 0 0 1 2 4 1 0 0 0 1 0 1 2 3 0 0 , 0 0 1 0 1 2 1 0 0 ,
18 6. If B is a square matrix, show that: a. T BB is symmetric. b. T B + B is symmetric. c. T B − B is skew-symmetric. 7. Solve the following non-linear system for x,y ,z and w. 2 2 2 3 1 1 1 1 1 1 1 1 5 1 1 1 + = − + + + = − − + = − + + = − x z x y z w x z w x z w 8. Solve the following systems of linear equations by Gauss-Jordan elimination: a. 2 4 2 9 3 5 3 3 15 + + = − + − = − + + = x y z x y z x y z b. 2 2 9 2 6 3 3 − + + − = + − = + − = x y z w x y z y z w c. 3 3 11 3 3 3 15 2 6 + + = − − = − + = x y z x y z y z d. 1 1 1 1 + = + = + = + = x w z w y z x y e. 7 7 0 2 5 2 0 0 − + + = − + + = + + = x y z x y z x y z f. 2 3 0 5 2 6 0 − + + = + + = x y z x y z g. 3 2 3 2 3 10 1 + + = − + + = − + − = x y z x y z x y z h. 2 6 2 2 2 3 3 2 3 0 + + − = + + = − + + − = x y z w x y z x y z w 9. Solve the system of linear equations represented by the following augmented matrices: a. − − − − − − − − − − − 1 1 1 0 1 2 0 1 1 1 0 3 0 1 1 1 1 6 0 1 0 1 1 0 1 0 1 1 1 1 b. − − − − − 0 0 2 0 2 0 0 0 0 0 0 1 1 0 1 1 1 5 6 1 1
19 10. Determine the conditions on k so that the following systems i. have infinitely many solutions ii. are inconsistent iii. have unique solutions a. 4 ( 14) 2 3 5 2 2 3 4 2 + + − = + − + = + − = x y k z k x y z x y z b. 2 3 1 3 4 2 2 + − = + + = + + = x y z x y z k x y kz c. 1 1 1 + + = + + = + + = x y kz x ky z kx y z d. 2 3 1 2 3 2 3 1 2 + + = − + + = + + = x y kz k x y z x y z 11. Determine the conditions on a,b and c so that the following systems are consistent: a. x y z c x z b x y z a + + = + = + + = 2 3 2 b. x y z c x y z b x y z a + + = + − = − + = 3 2 2 3 2 4 12. What are the value(s) of for which the following system is consistent? − + + = − − = − + − = x y z x y z x y z 16 2 2 2 2 2 3 4 3 13. Show that the following matrices are not row-equivalent: − 0 1 10 1 0 7 , 4 1 2 1 2 3 14. Show that − − 1 1 1 1 1 0 1 1 0 1 0 1 ~ 0 0 0 1 0 1 0 0 1 1 1 0 . 15. a. Explain the difference between the augmented matrix and the coefficient matrix for the system of linear equations given by Ax = b . b. Consider the following matrix: = 0 0 0 1 0 0 1 0 1 1 0 0 A i. If A is an augmented matrix representing a system of linear equations, find the solution set of the system. ii. If A is a coefficient matrix for a homogeneous system of linear equations, find the solution set of the system.
20 16. Find matrix A for each of the following expressions: a.( ) − − = − 1 2 3 7 7A 1 b. ( ) − − = 5 2 3 1 5A T c. ( ) + = − 1 1 2 0 I 2A 1 d. = − 1 2 2 3 0 1 1 -1 A 1 17. If − − = 5 2 1 1 A 3 , find ( ) 3 2A − . 18. Find the inverse of the following matrix: . sin cos cos sin − 19. Find the inverse of each of the following matrices using elementary row operations: a. − − 0 0 1 0 1 0 1 1 0 b. − − − − − 0 0 2 0 1 2 2 0 1 c. − − − − 1 6 1 1 2 1 1 2 1 d. − − − 1 1 1 1 0 1 1 1 0 e. 3 6 5 3 5 4 1 1 1 f. 3 3 1 3 2 1 1 3 2 20. If − = − 7 6 2 5 A 1 and − = − 2 0 7 3 B 1 , find (AB) ,(A ) ,A , 2 1 1 T − − − and ( ) 1 2A − . 21. Consider the following matrices: − − = − − = − − = 1 2 0 0 1 2 0 4 3 ,C 1 2 3 0 1 2 1 2 0 ,B 1 2 0 0 1 2 1 2 3 A a. Determine the elementary matrix E such that EA=B. b. Determine the elementary matrix E such that EA=C. c. Determine the elementary matrix E such that EB=A. d. Determine the elementary matrix E such that EC=A.
21 22. Consider the following matrices: = − − − = − − − 1 2 3 1 4 1 1 3 3 ,B 0 1 0 1 4 1 1 2 3 A Obtain elementary matrices E1 and E2 such that A = E2E1B. 23. a.What is an elementary matrix? Give an example each of an elementary matrix and a non-elementary matrix. b. If − = 0 0 2 5 1 3 1 4 A and − − = 0 0 0 3 1 3 1 4 B , find an elementary matrix E such that EA=B. Find also 1 E − . 24. Express the matrix − − − = 2 5 1 8 1 3 3 8 0 1 7 8 A in the form A=EFGR, where E,F and G are elementary matrices and R is a matrix in row-echelon form. 25. Express A as a product of elementary matrices: a. 1 3 2 4 A = − b. 1 2 1 2 5 3 1 2 0 A − = ================================================================================== Answer 1. a. 3 6 ; 11 4 8 AB BA = = b. 5 10 3 4 ; 4 8 12 16 AB BA undefined − = = c. 3 0 3 10 4 2 5 3 ; 5 1 7 4 3 AB BA − − − = − − = − − − −
22 d. 37 66 51 56 46 ; 9 11 17 15 19 AB undefined BA − = = − − − − 3. 1 1 1 201 4 2 3 − − − 5. Reduced row echelon form Neither Reduced row echelon form Row echelon form Neither Reduced row echelon form Neither Neither Neither Neither Row echelon form Neither Row echelon form 7. 1 1 2 1 2 1 4 x y z w = − −
23 9. a. 1 2 3 4 5 7 4 6 1 x t x t x x t x − = − − b. 1 2 3 4 5 6 1 5 6 , , , x r s t x r x t r s t x s x t x t + + − = 11. a. c a b a b c − − = 0 where , , b. a b c a b c + − 0 where , , 15. bi. No solution ii. 1 2 3 4 0 , 0 x p x p x p x − = 17. 1 1 12 24 5 1 24 24 − − 19. a. 1 1 0 0 1 0 0 0 1 − − − b. 1 1 0 2 4 0 1 1 1 0 0 2 − − − c. no inverse d. 1 1 1 0 1 1 1 2 1 − − − − − e. 1 1 1 3 2 1 3 3 2 − − − − f. 1 3 1 5 5 5 0 1 1 3 6 7 5 5 5 − − − − 21. a. 0 0 1 0 1 0 1 0 0 b. 1 0 1 0 1 0 0 0 1
24 c. 0 0 1 0 1 0 1 0 0 d. 1 0 1 0 1 0 0 0 1 − 23. b. 1 0 0 0 0 1 0 0 2 0 1 0 0 0 0 1 − , 1 0 0 0 0 1 0 0 2 0 1 0 0 0 0 1 25. a. 1 0 1 0 1 3 2 1 0 10 0 1 − b. 1 0 0 1 0 0 1 2 0 1 0 11 1 0 0 2 1 0 0 1 0 0 1 0 0 1 0 0 1 5 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 −
1 Exercise 1 1. Compute AB and BA (if defined): a. , B (1 2) 4 3 A = = b. − − = − = 2 4 3 0 1 2 3 4 , B 4 0 1 2 A c. = − − − = 0 1 1 1 0 1 , B 7 4 2 5 3 0 A d. − − = − − = 3 1 1 6 , B 6 11 8 9 7 1 0 3 2 4 A 2. Find 8 A if . 0 1 1 1 A = 3. Consider the following matrix: − − = 0 2 1 2 0 1 1 1 1 A Show that A A 3 = . Hence, find A . 40 4. Find A ,A A,AA : T T T a. = − 7 5 0 4 6 0 A b. − = 1 3 2 A 5. State whether each of the following matrices is in row-echelon form,reduced rowechelon form or neither. 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 3 0 3 0 0 1 0 5 0 0 1 5 1 0 0 5 , 0 1 3 3 1 0 3 1 , − 0 1 3 3 1 9 3 3 , 0 0 0 0 0 0 0 0 0 1 1 0 3 3 0 1 3 0 3 0 0 0 0 0 0 0 0 0 0 , 1 0 0 1 , − 0 1 2 1 , 0 1 0 0 0 1 1 0 0 , 0 0 1 1 0 0 0 0 0 , 0 0 0 0 1 0 0 1 0 0 1 2 4 1 0 0 0 1 0 1 2 3 0 0 , 0 0 1 0 1 2 1 0 0 , 6. If B is a square matrix, show that: a. T BB is symmetric. b. T B + B is symmetric. c. T B − B is skew-symmetric.
2 7. Solve the following non-linear system for x,y ,z and w. 2 2 2 3 1 1 1 1 1 1 1 1 5 1 1 1 + = − + + + = − − + = − + + = − x z x y z w x z w x z w 8. Solve the following systems of linear equations by Gauss-Jordan elimination: a. 2 4 2 9 3 5 3 3 15 + + = − + − = − + + = x y z x y z x y z b. 2 2 9 2 6 3 3 − + + − = + − = + − = x y z w x y z y z w c. 3 3 11 3 3 3 15 2 6 + + = − − = − + = x y z x y z y z d. 1 1 1 1 + = + = + = + = x w z w y z x y e. 7 7 0 2 5 2 0 0 − + + = − + + = + + = x y z x y z x y z f. 2 3 0 5 2 6 0 − + + = + + = x y z x y z g. 3 2 3 2 3 10 1 + + = − + + = − + − = x y z x y z x y z h. 2 6 2 2 2 3 3 2 3 0 + + − = + + = − + + − = x y z w x y z x y z w 9. Solve the system of linear equations represented by the following augmented matrices: a. − − − − − − − − − − − 1 1 1 0 1 2 0 1 1 1 0 3 0 1 1 1 1 6 0 1 0 1 1 0 1 0 1 1 1 1 b. − − − − − 0 0 2 0 2 0 0 0 0 0 0 1 1 0 1 1 1 5 6 1 1
3 10. Determine the conditions on k so that the following systems i. have infinitely many solutions ii. are inconsistent iii. have unique solutions a. 4 ( 14) 2 3 5 2 2 3 4 2 + + − = + − + = + − = x y k z k x y z x y z b. 2 3 1 3 4 2 2 + − = + + = + + = x y z x y z k x y kz c. 1 1 1 + + = + + = + + = x y kz x ky z kx y z d. 2 3 1 2 3 2 3 1 2 + + = − + + = + + = x y kz k x y z x y z 11. Determine the conditions on a,b and c so that the following systems are consistent: a. x y z c x z b x y z a + + = + = + + = 2 3 2 b. x y z c x y z b x y z a + + = + − = − + = 3 2 2 3 2 4 12. What are the value(s) of for which the following system is consistent? − + + = − − = − + − = x y z x y z x y z 16 2 2 2 2 2 3 4 3 13. Show that the following matrices are not row-equivalent: − 0 1 10 1 0 7 , 4 1 2 1 2 3 14. Show that − − 1 1 1 1 1 0 1 1 0 1 0 1 ~ 0 0 0 1 0 1 0 0 1 1 1 0 15. a. Explain the difference between the augmented matrix and the coefficient matrix for the system of linear equations given by Ax = b . b. Consider the following matrix: = 0 0 0 1 0 0 1 0 1 1 0 0 A i. If A is an augmented matrix representing a system of linear equations, find the solution set of the system. ii. If A is a coefficient matrix for a homogeneous system of linear equations, find the solution set of the system.
4 16. Find matrix A for each of the following expressions: a. ( ) − − = − 1 2 3 7 7A 1 b. ( ) − − = 5 2 3 1 5A T c. ( ) + = − 1 1 2 0 I 2A 1 d. = − 1 2 2 3 0 1 1 -1 A 1 17. If − − = 5 2 1 1 A 3 , find ( ) 3 2A − . 18. Find the inverse of the following matrix: . sin cos cos sin − 19. Find the inverse of each of the following matrices using elementary row operations: a. − − 0 0 1 0 1 0 1 1 0 b. − − − − − 0 0 2 0 1 2 2 0 1 c. − − − − 1 6 1 1 2 1 1 2 1 d. − − − 1 1 1 1 0 1 1 1 0 e. 3 6 5 3 5 4 1 1 1 f. 3 3 1 3 2 1 1 3 2 20. If − = − 7 6 2 5 A 1 and − = − 2 0 7 3 B 1 , find (AB) ,(A ) ,A , 2 1 1 T − − − and (2A) , −1 . 21. Consider the following matrices: − − = − − = − − = 1 2 0 0 1 2 0 4 3 ,C 1 2 3 0 1 2 1 2 0 ,B 1 2 0 0 1 2 1 2 3 A a. Determine the elementary matrix E such that EA=B. b. Determine the elementary matrix E such that EA=C. c. Determine the elementary matrix E such that EB=A. d. Determine the elementary matrix E such that EC=A.
5 22. Consider the following matrices: = − − − = − − − 1 2 3 1 4 1 1 3 3 ,B 0 1 0 1 4 1 1 2 3 A Obtain elementary matrices E1 and E2 such that A = E2E1B. 23. a.What is an elementary matrix? Give an example each of an elementary matrix and a non-elementary matrix. b. If − = 0 0 2 5 1 3 1 4 A and − − = 0 0 0 3 1 3 1 4 B , find an elementary matrix E such that EA=B. Find also 1 E − . 24. Express the matrix − − − = 2 5 1 8 1 3 3 8 0 1 7 8 A in the form A=EFGR, where E,F and G are elementary matrices and R is a matrix in row-echelon form. 25. Express A as a product of elementary matrices: a. 1 3 2 4 A = − b. 1 2 1 2 5 3 1 2 0 A − = Answer 1. a. 3 6 ; 11 4 8 AB BA = = b. 5 10 3 4 ; 4 8 12 16 AB BA undefined − = = c. 3 0 3 10 4 2 5 3 ; 5 1 7 4 3 AB BA − − − = − − = − − − − d. 37 66 51 56 46 ; 9 11 17 15 19 AB undefined BA − = = − − − −
6 3. 1 1 1 201 4 2 3 − − − 5. Reduced row echelon form Neither Reduced row echelon form Row echelon form Neither Reduced row echelon form Neither Neither Neither Neither Row echelon form Neither Row echelon form 7. 1 1 2 1 2 1 4 x y z w = − − 9. a. 1 2 3 4 5 7 4 6 1 x t x t x x t x − = − − b. 1 2 3 4 5 6 1 5 6 , , , x r s t x r x t r s t x s x t x t + + − = 11. a. c a b a b c − − = 0 where , , b. a b c a b c + − 0 where , ,
7 15. bi. No solution ii. 1 2 3 4 0 , 0 x p x p x p x − = 17. 1 1 12 24 5 1 24 24 − − 19. a. 1 1 0 0 1 0 0 0 1 − − − b. 1 1 0 2 4 0 1 1 1 0 0 2 − − − c. no inverse d. 1 1 1 0 1 1 1 2 1 − − − − − e. 1 1 1 3 2 1 3 3 2 − − − − f. 1 3 1 5 5 5 0 1 1 3 6 7 5 5 5 − − − − 21. a. 0 0 1 0 1 0 1 0 0 b. 1 0 1 0 1 0 0 0 1 c. 0 0 1 0 1 0 1 0 0 d. 1 0 1 0 1 0 0 0 1 − 23. b. 1 0 0 0 0 1 0 0 2 0 1 0 0 0 0 1 − , 1 0 0 0 0 1 0 0 2 0 1 0 0 0 0 1 25. a. 1 0 1 0 1 3 2 1 0 10 0 1 −
8 b. 1 0 0 1 0 0 1 2 0 1 0 11 1 0 0 2 1 0 0 1 0 0 1 0 0 1 0 0 1 5 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 −
1 CHAPTER 2 DETERMINANTS 2.1 Symbol: A or det(A) a. = ad − bc c d a b b. a(ei fh) b(di fg) c(dh eg) g h i d e f a b c = − − − + − Example 1 Evaluate the determinant of each of the following matrices: a. − = 4 2 3 1 A b. − − = 2 2 1 4 B c. − = − 7 8 9 4 5 6 1 2 3 C d. = 1 2 2 2 1 2 2 1 3 D e. − − − = 3 6 1 1 4 2 3 1 2 F f. − − − = 2 8 4 3 3 1 1 4 2 G g. n I
2 2.2 Properties of Determinants Suppose A and B are both square matrices. 1. If A has a row (column) of zeroes, then A = 0 . 2. T A = A . 3. If A is nxn triangular matrix, then A is the product of the entries on the main diagonal, i.e. nn A = a11a22 a . 4. AB = A B . 5. If A has two equal or proportional rows (columns) then A = 0 . 6. If B is the matrix that results when a single row (column) of A is multiplied by a scalar k, then B = k A . [If A and B are nxn matrices and B=kA, then B k A n = because every row has been multiplied by k] 7. If B is the matrix that results when two rows (columns) of A are interchanged, then, B = − A . 8. If B is the matrix that results when a multiple of one row (column) of A is added to another row (column), then B = A .
3 Example 2 Compute the following determinants. a. 0 0 0 0 4 0 0 0 9 1 0 0 3 1 2 0 5 6 0 7 2 7 3 8 3 − − − b. dn d d 0 0 0 0 0 0 0 0 0 2 1 (diagonal matrix) Example 3 Compute the following determinants. a. 1 2 1 0 3 6 1 2 1 b. 2 4 2 0 3 6 1 2 1 − − − Example 4 B 1 2 1 0 1 4 1 2 1 0 1 4 1 2 3 A R1 4R1 = ⎯⎯ ⎯→ = → Find A and B .
4 Example 5 0 1 4 B 1 2 1 0 1 4 1 2 3 A R1 R3 = ⎯⎯⎯→ = Find A and B . Example 6 B 1 2 1 1 2 3 1 2 1 0 1 4 1 2 3 A R2 R2 2R3 = ⎯⎯⎯ ⎯→ = → − Find A and B .
5 Further Examples 1. Use the properties of determinants to compute the following determinants. a. 1003 1004 1005 1001 1002 1003 999 1000 1001 b. c a b b a c a b c + + + 1 1 1 2. If = 5 g h i d e f a b c , evaluate a b c d a e b f c g h i − − − − − − 2 2 2 3 3 3 . 3. If 3 2 2 2 = + + + − − − a g b h c i g h i d e f , evaluate g h i d e f a b c . 4. If 8 2 2 2 − − − = g h i d e f a b c , evaluate g h i d e f a d b e c f 3 3 3 + 2 + 2 + 2 .
6 2.3 Determinants of Elementary Matrices Theorem Let E be an nxn elementary matrix. a. If E is obtained from n I by multiplying a row by k, then E = k . b. If E is obtained from n I by interchanging 2 rows, then E = -1. c. If E is obtained from n I by adding the multiple of one row to another, then E = 1. Example 7 Compute the following determinants. a. 0 0 0 1 0 0 1 0 0 3 0 0 1 0 0 0 b. 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 c. 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 7 Theorem A square matrix, A is invertible if and only if A 0. Theorem If A is invertible then . A 1 A -1 = Theorem If A is an nxn matrix, then the following are equivalent: a. A is invertible. b. Ax = 0 has only the trivial solution. c. The reduced row-echelon form of A is n I . d. A is expressible as a product of elementary matrices. e. Ax = b is consistent. f. Ax = b has a unique solution. g. A 0.
7 2.4 Cofactor Expansion; Cramer’s Rule Definition If A is a square matrix, then the minor of entry ij a is denoted by Mij and is defined to be the determinant of the sub-matrix that remains after the i-th row and the j-th column are deleted from A. The cofactor of entry ij a is denoted by Cij and given by ( ) ij i j Cij 1 M + = − . Example 8 If − = 1 4 8 2 5 6 3 1 4 A then the minors and cofactors of A are: = = = = = = = = = = = = = = = = = = 3 1 3 2 3 3 3 1 3 2 3 3 2 1 2 2 2 3 2 1 2 2 2 3 1 1 1 2 1 3 1 1 1 2 1 3 M M M C C C M M M C C C M 16 M M C C C
8 Consider the following 3x3 matrix: 31 32 33 21 22 23 11 12 13 a a a a a a a a a ( ) ( ) ( ) 1 1 1 1 1 2 1 2 1 3 1 3 1 1 2 2 3 3 2 3 3 2 1 2 2 1 3 3 2 3 3 1 1 3 2 1 3 2 2 2 3 1 C C C A a a a a a a a a a a a a a a a a a a = + + = − − − + − This method of evaluating A is called cofactor expansion along the first row of A. Expansion can be done along any row or column of A. For a 3x3 matrix, A = a11C11 + a12C12 + a13C13 = a21C21 + a22C22 + a23C23 = a31C31 + a32C32 + a33C33 = a11C11 + a21C21 + a31C31 = a12C12 + a22C22 + a32C32 = a13C13 + a23C23 + a33C33
9 Example 9 If − = − − 5 4 2 2 4 3 3 1 0 A , evaluate A by cofactor expansion along the first column of A.
10 Example 10 Compute the following determinant by cofactor expansion along the third column. 2 3 1 4 1 7 0 3 3 4 2 2 5 1 2 1 − − − −