Creating matrix and edit matrix Operation of matrix Mode matrices Determinant of matrix Transposition of matrix Inverse matrix MATRICES 44
MATRICES • The element of matrix A is denoted as , where i and j is the number of row and column of matrix A respectively. • The order of matrix A is defined by × , where m and n is the total number of row and column of matrix A respectively. 31 32 33 21 22 23 11 12 13 a a a a a a a a a A MATRIX (3X3) COMMONLY WRITTEN AS: Column 1 Column 2 Column 3 Row 1 Row 2 Row 3 45
MODE OF MATRIX Press MODE button 3 time you will find MAT mode at second option, then 2 press . EQN MAT VCT2 1 2 3 MODE 46
MATRIX CALCULATIONS Press (SHIFT) then (MAT) , you will find this three option. Dim Edit Mat Shift 1 2 3 MAT Dim 1 Edit 2 Mat 3 Used to creating the matrices with define matrix dimension first and then input the element of matrices. Used to edit the matrices Used to display the matrix for operation matrices 47
MATRIX CALCULATIONS Press (SHIFT) (MAT) , you will find this two option. Det Trn Shift 1 2 MAT Det 1 Trn 2 Used to find the determinant of matrices. Used to transpose the matrices 48
CREATING A MATRIX 4 2 0 7 1 5 A MatA(mxn) m ? MatA(mxn) n ? MatA11 MatA21 MatA31 MatA12 MatA22 MatA32 ROWS: 3 COLUMNS: 2 ORDER OF MATRICES: 3x2 SHIFT MAT Dim 1 MATRIX NAME A(1) DIMENSION m : no. rows ; n: no. columns ELEMENT OF MATRIX Example: 49
EDIT A MATRIX Change the element of A12 =6 MatA12 Use the cursor keys to move about the matrix in order to edit its elements. SHIFT MAT Edit 2 MATRIX NAME A(1) ELEMENT OF MATRIX Example: 50
OPERATIONS OF MATRIX: ADDITION OF MATRICES Find the sum of following matrix below: 3 2 4 1 1 4 4 2 0 7 1 6 A B Remember that two matrix can be add or subtracted only if they have the same order. Example: 51
CREATING A MATRIX MatB(mxn) m ? MatB(mxn) n ? MatB11 MatB21 MatB31 MatB12 MatB22 MatB32 ROWS: 3 COLUMNS: 2 ORDER OF MATRICES: 3 x2 SHIFT MAT Dim 1 MATRIX NAME B(2) DIMENSION m : no. rows ; n: no. columns ELEMENT OF MATRIX 3 2 4 1 1 4 B 52
Mat A + Mat B Press SHIFT MAT Mat 3 MATRIX NAME A(1) DISPLAY Mat A SHIFT MAT Mat 3 MATRIX NAME B(2) DISPLAY Mat B OPERATOR DISPLAY MatAns11 = 2 MatAns21 = 4 MatAns31 =-7 MatAns12 = 2 MatAns22 = 8 MatAns32 = 4 ANSWERS Use your cursor keys to move in order to view each elements for the answers. 7 4 4 8 2 2 53
OPERATIONS OF MATRIX: SUBTRACTION OF MATRICES Find the subtraction of following matrix below: 3 2 4 1 1 4 4 2 0 7 1 6 A B When use the same element of matrices, then no need to create the new matrix. Example: 54
Mat A - Mat B Press SHIFT MAT Mat 3 MATRIX NAME A(1) DISPLAY Mat A SHIFT MAT Mat 3 MATRIX NAME B(2) DISPLAY Mat B OPERATOR DISPLAY MatAns11 = 0 MatAns21 = -4 MatAns31 =-1 MatAns12 = 10 MatAns22 = 6 MatAns32 = 0 ANSWERS Use your cursor keys to move in order to view each elements for the answers. 0 10 −4 6 −1 0 55
OPERATIONS OF MATRIX: MULTIPLICATION OF MATRICES BY SCALAR NUMBER Given that , then evaluate × 1 2 3 4 2 5 C When use the different element of matrices, then create the new matrix. Example: 56
CREATING A MATRIX MatC(mxn) m ? MatC(mxn) n ? MatC11 MatC13 MatC22 MatC12 MatC21 MatC23 ROWS: 2 COLUMNS: 3 ORDER OF MATRICES: 2x3 SHIFT MAT Dim 1 MATRIX NAME C(3) DIMENSION m : no. rows ; n: no. columns ELEMENT OF MATRIX 1 2 3 4 2 5 C 57
2 x Mat C Then, press Press the scalar number SHIFT MAT Mat 3 MATRIX NAME C(3) DISPLAY Mat C OPERATOR DISPLAY MatAns11 = 8 MatAns21 = -2 MatAns12 = 4 MatAns22 = -4 ANSWERS MatAns13 = 10 MatAns23 = 6 Use your cursor keys to move in order to view each elements for the answers. 2 4 6 8 4 10 58
MULTIPLICATION OF TWO MATRICES Find the multiplication of following matrix below: 3 2 4 1 1 4 B C 1 2 3 4 2 5 same Two matrix B & matrix C can be multiplied only if the number of columns of matrix B is same as the number of rows of matrix C. The order of new matrix after multiplied matrix B and C equal to the number of row of matrix B x number of column of matrix C. same 3 X 2 2 X 3 3 X 3 Example: 59
Press OPERATOR SHIFT MAT Mat 3 MATRIX NAME C(3) DISPLAY Mat C SHIFT MAT Mat 3 MATRIX NAME B(2) DISPLAY Mat B Mat B x Mat C DISPLAY ANSWERS 14 10 9 15 6 23 8 10 7 MatAns11 = 8 MatAns21 = 15 MatAns31 = -14 MatAns12 = 10 MatAns22 = 6 MatAns32 =-1 0 MatAns12 = -7 MatAns22 = 23 MatAns32 = -9 Use your cursor keys to move in order to view each elements for the answers. 60
DETERMINANT OF MATRICES (2X2) Determinant of matrix can be computed from the elements of a square matrix. Find the determinant of the matrix When use the different element of matrices, then create the new matrix. M= 2 −1 4 3 Example: 61
CREATING A MATRIX MatA(mxn) m ? MatA(mxn) n ? MatA11 MatA21 MatA12 MatA22 ROWS: 2 COLUMNS: 2 ORDER OF MATRICES: 2x2 SHIFT MAT Dim 1 MATRIX NAME A(1) DIMENSION m : no. rows ; n: no. columns ELEMENT OF MATRIX M= 2 −1 4 3 62
DetMat A SHIFT MAT CURSOR KEYS Det 1 DISPLAY Det SHIFT MAT Mat 3 MATRIX NAME A(1) DISPLAY Mat A DISPLAY 10 4 3 2 1 det ANSWERS Answer: =10 Press 63
DETERMINANT OF MATRICES (3X3) Determinant of matrix can be computed from the elements of a square matrix. Find the determinant of the matrix . 2 2 0 4 1 3 1 3 2 B When use the different element of matrices, then create the new matrix. Example: 64
CREATING A MATRIX MatB(mxn) m? MatB(mxn) n? MatB11 MatB13 MatB22 MatB12 MatB21 MatB23 SHIFT MAT Dim 1 MATRIX NAME B(2) DIMENSION m : no. rows ; n: no. columns ELEMENT OF MATRIX 2 2 0 4 1 3 1 3 2 B MatB31 MatB32 MatB33 ROWS: 3 COLUMNS: 3 ORDER OF MATRICES: 3x3 65
SHIFT MAT CURSOR KEYS Det 1 DISPLAY Det SHIFT MAT Mat 3 MATRIX NAME B(2) DISPLAY Mat B DetMat B DISPLAY 24 ANSWERS Answer: =24 2 2 0 4 1 3 1 3 2 det 66
TRANSPOSITION OF MATRICES Simplify the matrix below: B T 2 2 0 4 1 3 1 3 2 When use the same element of matrices, then no need to create the new matrix. Example: 67
TrnMat B SHIFT MAT CURSOR KEYS Trn DISPLAY 2 DISPLAY Trn SHIFT MAT Mat 3 MATRIX NAME B(2) DISPLAY Mat B Press MatAns11 = 1 MatAns21 = 3 MatAns12 =4 MatAns22 =1 ANSWERS MatAns13 = 2 MatAns23 = 2 MatAns31 = 2 MatAns32 = 3 MatAns33 = 0 Use your cursor keys to move in order to view each elements for the answers. 2 3 0 3 1 2 1 4 2 T B 68
INVERSE OF MATRICES (2X2) For the matrix , find the inverse matrix: A When use the same element of matrices, then no need to create the new matrix. 4 3 2 1 A Example: 69
Press 1 A DISPLAY SHIFT MAT Mat 3 MATRIX NAME A(1) DISPLAY Mat A 1 x 1 Mat A MatAns11 = 3/10 MatAns21 = -2/5 MatAns12 =1/10 MatAns22 = 1/5 ANSWERS 5 1 5 2 10 1 10 3 Convert your answer from decimal places into fraction with press Then, use your cursor keys to move in order to view each elements for the answers. 70
INVERSE OF MATRICES (3X3) For the matrix , find the inverse matrix: B When use the same element of matrices, then no need to create the new matrix. 2 2 0 4 1 3 1 3 2 B Example: 71
Press 1 B 1 MatB SHIFT MAT Mat 3 MATRIX NAME B(2) DISPLAY Mat B 1 x DISPLAY Convert your answer from decimal places into fraction with press Then, use your cursor keys to move in order to view each elements for the answers. MatAns11 = -1/4 MatAns21 = 1/4 MatAns12 =1/6 MatAns22 =- 1/6 ANSWERS MatAns13 = 7/24 MatAns23 = 5/24 MatAns31 = 1/4 MatAns32 = 1/6 MatAns33 = -11/24 − 1 4 1 6 7 24 1 4 − 1 6 5 24 1 4 1 6 − 11 24 1 B = 72
CALCULATOR’S RESET 73
Creating vector and edit vector Operation of vector Mode vector Scalar (dot) product of two vectors Vector (cross) product of two vectors Magnitude of vector and unit vector VECTOR 74
MODE OF VECTOR Press MODE button 3 time you will find VCT mode at third option, then 3 press . EQN MAT VCT 1 2 3 MODE 75
VECTOR CALCULATION Press (SHIFT) then (VCT) you will find this at three option. Dim Edit Vct Shift 1 2 3 VCT Dim 1 Edit 2 Vct 3 Used to creating the vector with define vector dimension first and then input the element of the vector. Used to edit the element of the vector. Used to display the vector for operation vector. 76
VECTOR CALCULATION Press (SHIFT) (VCT) (right cursor keys) , you will find this at one option. Dot Shift 1 VCT Dot 1 Dot operation which is used to solve the dot product of vector. 77
CREATING A VECTOR 1 5 1 A Vct A(m) m ? SHIFT VCT Dim 1 VECTOR NAME A(1) DIMENSION m : Dimension of Vector ELEMENT OF VECTOR VctA1 VctA2 VctA3 Dimension of Example: vector: 3 78
EDIT A VECTOR Change the element of A2 =3 VctA2 Use the cursor keys to move about the matrix in order to edit its elements. SHIFT VCT Edit 2 VECTOR NAME A(1) ELEMENT OF MATRIX Example: 79
OPERATIONS OF VECTOR: ADDITION OF VECTOR Given that and , find 5i 2 j 4k 3i 2 j 6k N M M N Remember that two vector can be add or subtracted only if they have the same dimension. M N 3i 2 j 6k 5i 2 j 4k 80
CREATING A VECTOR Vct A(m) m ? SHIFT VCT Dim 1 VECTOR NAME A(1) DIMENSION m : Dimension of Vector ELEMENT OF VECTOR VctA1 VctA2 VctA3 3i 2 j 6k M Dimension of vector: 3 Example: 81
CREATING A VECTOR Vct B(m) m ? SHIFT VCT Dim 1 VECTOR NAME B(2) DIMENSION m : Dimension of Vector ELEMENT OF VECTOR VctB1 VctB2 VctB3 5i 2 j 4k N Dimension of Example: vector: 3 82
Press OPERATOR SHIFT VCT Vct 3 VECTOR NAME B(2) DISPLAY VctB SHIFT VCT Vct 3 VECTOR NAME A(1) DISPLAY VctA Vct A + Vct B DISPLAY Vct Ans1 = 8 Vct Ans2 = -4 Vct Ans3 =2 ANSWERS 8i 4 j 2k M N Use your cursor keys to move in order to view each elements for the answers. 83
Given that and , find A B Remember that two vector can be add or subtracted only if they have the same dimension. M N 3i 2 j 6k 5i 2 j 4k 5i 2 j 4k 3i 2 j 6k N M OPERATIONS OF VECTOR: SUBTRACTION OF VECTOR When use the same element of vector, then no need to create the new vector. Example: 84
Press Vct A - Vct B DISPLAY OPERATOR SHIFT VCT Vct 3 VECTOR NAME B(2) DISPLAY VctB SHIFT VCT Vct 3 VECTOR NAME A(1) DISPLAY VctA ANSWERS Vct Ans1 = -2 Vct Ans2 = 0 Vct Ans3 = 10 2i 10k M N Use your cursor keys to move in order to view each elements for the answers. 85
OPERATIONS OF VECTOR: MULTIPLICATION OF VECTOR BY SCALAR NUMBER Given that , then evaluate 3, 2 C When use the different element of vector, then create the new vector. 2 × Ԧ Example: 86
CREATING A VECTOR Vct C(m) m ? SHIFT VCT Dim 1 VECTOR NAME C(3) DIMENSION m : Dimension of Vector ELEMENT OF VECTOR VctC1 VctC2 3, 2 C Dimension of vector: 2 87
2 x VctC Then, press Press the scalar number SHIFT VCT Vct 3 MATRIX NAME C(3) DISPLAY VctC OPERATOR DISPLAY ANSWERS Vct Ans1 = 6 Vct Ans2 = - 4 2 × Ԧ = , − @ − Use your cursor keys to move in order to view each elements for the answers. 88
Given that and , find SCALAR (DOT) PRODUCT OF TWO VECTORS OPOQ 5i 4 j 4k 9i 3 j 8k OQ OP Remember that two vector can be apply dot product only if they have the same dimension. When use the different vector, then create the new vector. A B 9i 3 j 8k 5i 4 j 4k Example: 89
CREATING A VECTOR Vct A(m) m ? SHIFT VCT Dim 1 VECTOR NAME A(1) DIMENSION m : Dimension of Vector ELEMENT OF VECTOR Dimension of k vector: 3 9i 3 j 8 OP VctA1 VctA2 VctA3 90
CREATING A VECTOR Vct B(m) m ? SHIFT VCT Dim 1 VECTOR NAME B(2) DIMENSION m : Dimension of Vector ELEMENT OF VECTOR Dimension of k vector: 3 5i 4 j 4 OQ VctB1 VctB2 VctB3 91
Press Vct A Vct B DISPLAY Dot 1 SHIFT VCT Vct 3 VECTOR NAME B(2) DISPLAY VctB SHIFT VCT Vct 3 VECTOR NAME A(1) DISPLAY VctA SHIFT VCT CURSOR KEYS OPERATOR -25 25 OP OQ Answer: 92
Given that and , find VECTOR (CROSS) PRODUCT OF TWO VECTORS OPOQ 5i 4 j 4k 9i 3 j 8k OQ OP A B 9i 3 j 8k 5i 4 j 4k Remember that two vector can be apply cross product only if they have the same dimension. When use the same element of vector, then no need to create the new vector. Example: 93