Here are the calculations for C = A B C=AB C = A B
a) Find C = A B C=AB C = A B
Given:
A = [ 12 14 20 5 ] , B = [ 3 9 0 2 ] A = \begin{bmatrix} 12 & 14 \\ 20 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 9 \\ 0 & 2 \end{bmatrix} A = [ 12 20 14 5 ] , B = [ 3 0 9 2 ] C = A B C=AB C = A B C = [ 12 14 20 5 ] [ 3 9 0 2 ] C = \begin{bmatrix} 12 & 14 \\ 20 & 5 \end{bmatrix} \begin{bmatrix} 3 & 9 \\ 0 & 2 \end{bmatrix} C = [ 12 20 14 5 ] [ 3 0 9 2 ] C = [ ( 12 ) ( 3 ) + ( 14 ) ( 0 ) ( 12 ) ( 9 ) + ( 14 ) ( 2 ) ( 20 ) ( 3 ) + ( 5 ) ( 0 ) ( 20 ) ( 9 ) + ( 5 ) ( 2 ) ] C = \begin{bmatrix} (12)(3)+(14)(0) & (12)(9)+(14)(2) \\ (20)(3)+(5)(0) & (20)(9)+(5)(2) \end{bmatrix} C = [ ( 12 ) ( 3 ) + ( 14 ) ( 0 ) ( 20 ) ( 3 ) + ( 5 ) ( 0 ) ( 12 ) ( 9 ) + ( 14 ) ( 2 ) ( 20 ) ( 9 ) + ( 5 ) ( 2 ) ] C = [ 36 + 0 108 + 28 60 + 0 180 + 10 ] C = \begin{bmatrix} 36+0 & 108+28 \\ 60+0 & 180+10 \end{bmatrix} C = [ 36 + 0 60 + 0 108 + 28 180 + 10 ] C = [ 36 136 60 190 ] C = \begin{bmatrix} 36 & 136 \\ 60 & 190 \end{bmatrix} C = [ 36 60 136 190 ] C = [ 36 136 60 190 ] \boxed{C = \begin{bmatrix} 36 & 136 \\ 60 & 190 \end{bmatrix}} C = [ 36 60 136 190 ]
b) Find C = A B C=AB C = A B
Given:
A = [ 4 7 9 1 ] , B = [ 3 8 5 2 6 7 ] A = \begin{bmatrix} 4 & 7 \\ 9 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 8 & 5 \\ 2 & 6 & 7 \end{bmatrix} A = [ 4 9 7 1 ] , B = [ 3 2 8 6 5 7 ] C = A B C=AB C = A B C = [ 4 7 9 1 ] [ 3 8 5 2 6 7 ] C = \begin{bmatrix} 4 & 7 \\ 9 & 1 \end{bmatrix} \begin{bmatrix} 3 & 8 & 5 \\ 2 & 6 & 7 \end{bmatrix} C = [ 4 9 7 1 ] [ 3 2 8 6 5 7 ] C = [ ( 4 ) ( 3 ) + ( 7 ) ( 2 ) ( 4 ) ( 8 ) + ( 7 ) ( 6 ) ( 4 ) ( 5 ) + ( 7 ) ( 7 ) ( 9 ) ( 3 ) + ( 1 ) ( 2 ) ( 9 ) ( 8 ) + ( 1 ) ( 6 ) ( 9 ) ( 5 ) + ( 1 ) ( 7 ) ] C = \begin{bmatrix} (4)(3)+(7)(2) & (4)(8)+(7)(6) & (4)(5)+(7)(7) \\ (9)(3)+(1)(2) & (9)(8)+(1)(6) & (9)(5)+(1)(7) \end{bmatrix} C = [ ( 4 ) ( 3 ) + ( 7 ) ( 2 ) ( 9 ) ( 3 ) + ( 1 ) ( 2 ) ( 4 ) ( 8 ) + ( 7 ) ( 6 ) ( 9 ) ( 8 ) + ( 1 ) ( 6 ) ( 4 ) ( 5 ) + ( 7 ) ( 7 ) ( 9 ) ( 5 ) + ( 1 ) ( 7 ) ] C = [ 12 + 14 32 + 42 20 + 49 27 + 2 72 + 6 45 + 7 ] C = \begin{bmatrix} 12+14 & 32+42 & 20+49 \\ 27+2 & 72+6 & 45+7 \end{bmatrix} C = [ 12 + 14 27 + 2 32 + 42 72 + 6 20 + 49 45 + 7 ] C = [ 26 74 69 29 78 52 ] C = \begin{bmatrix} 26 & 74 & 69 \\ 29 & 78 & 52 \end{bmatrix} C = [ 26 29 74 78 69 52 ] C = [ 26 74 69 29 78 52 ] \boxed{C = \begin{bmatrix} 26 & 74 & 69 \\ 29 & 78 & 52 \end{bmatrix}} C = [ 26 29 74 78 69 52 ]
c) Find C = A B C=AB C = A B
Given:
A = [ 7 11 2 9 10 6 ] , B = [ 12 4 5 3 6 1 ] A = \begin{bmatrix} 7 & 11 \\ 2 & 9 \\ 10 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 12 & 4 & 5 \\ 3 & 6 & 1 \end{bmatrix} A = 7 2 10 11 9 6 , B = [ 12 3 4 6 5 1 ] C = A B C=AB C = A B C = [ 7 11 2 9 10 6 ] [ 12 4 5 3 6 1 ] C = \begin{bmatrix} 7 & 11 \\ 2 & 9 \\ 10 & 6 \end{bmatrix} \begin{bmatrix} 12 & 4 & 5 \\ 3 & 6 & 1 \end{bmatrix} C = 7 2 10 11 9 6 [ 12 3 4 6 5 1 ] C = [ ( 7 ) ( 12 ) + ( 11 ) ( 3 ) ( 7 ) ( 4 ) + ( 11 ) ( 6 ) ( 7 ) ( 5 ) + ( 11 ) ( 1 ) ( 2 ) ( 12 ) + ( 9 ) ( 3 ) ( 2 ) ( 4 ) + ( 9 ) ( 6 ) ( 2 ) ( 5 ) + ( 9 ) ( 1 ) ( 10 ) ( 12 ) + ( 6 ) ( 3 ) ( 10 ) ( 4 ) + ( 6 ) ( 6 ) ( 10 ) ( 5 ) + ( 6 ) ( 1 ) ] C = \begin{bmatrix} (7)(12)+(11)(3) & (7)(4)+(11)(6) & (7)(5)+(11)(1) \\ (2)(12)+(9)(3) & (2)(4)+(9)(6) & (2)(5)+(9)(1) \\ (10)(12)+(6)(3) & (10)(4)+(6)(6) & (10)(5)+(6)(1) \end{bmatrix} C = ( 7 ) ( 12 ) + ( 11 ) ( 3 ) ( 2 ) ( 12 ) + ( 9 ) ( 3 ) ( 10 ) ( 12 ) + ( 6 ) ( 3 ) ( 7 ) ( 4 ) + ( 11 ) ( 6 ) ( 2 ) ( 4 ) + ( 9 ) ( 6 ) ( 10 ) ( 4 ) + ( 6 ) ( 6 ) ( 7 ) ( 5 ) + ( 11 ) ( 1 ) ( 2 ) ( 5 ) + ( 9 ) ( 1 ) ( 10 ) ( 5 ) + ( 6 ) ( 1 ) C = [ 84 + 33 28 + 66 35 + 11 24 + 27 8 + 54 10 + 9 120 + 18 40 + 36 50 + 6 ] C = \begin{bmatrix} 84+33 & 28+66 & 35+11 \\ 24+27 & 8+54 & 10+9 \\ 120+18 & 40+36 & 50+6 \end{bmatrix} C = 84 + 33 24 + 27 120 + 18 28 + 66 8 + 54 40 + 36 35 + 11 10 + 9 50 + 6 C = [ 117 94 46 51 62 19 138 76 56 ] C = \begin{bmatrix} 117 & 94 & 46 \\ 51 & 62 & 19 \\ 138 & 76 & 56 \end{bmatrix} C = 117 51 138 94 62 76 46 19 56 C = [ 117 94 46 51 62 19 138 76 56 ] \boxed{C = \begin{bmatrix} 117 & 94 & 46 \\ 51 & 62 & 19 \\ 138 & 76 & 56 \end{bmatrix}} C = 117 51 138 94 62 76 46 19 56
d) Find C = A B C=AB C = A B
Given:
A = [ 6 2 5 7 9 4 ] , B = [ 10 1 11 3 2 9 ] A = \begin{bmatrix} 6 & 2 & 5 \\ 7 & 9 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 10 & 1 \\ 11 & 3 \\ 2 & 9 \end{bmatrix} A = [ 6 7 2 9 5 4 ] , B = 10 11 2 1 3 9 C = A B C=AB C = A B C = [ 6 2 5 7 9 4 ] [ 10 1 11 3 2 9 ] C = \begin{bmatrix} 6 & 2 & 5 \\ 7 & 9 & 4 \end{bmatrix} \begin{bmatrix} 10 & 1 \\ 11 & 3 \\ 2 & 9 \end{bmatrix} C = [ 6 7 2 9 5 4 ] 10 11 2 1 3 9 C = [ ( 6 ) ( 10 ) + ( 2 ) ( 11 ) + ( 5 ) ( 2 ) ( 6 ) ( 1 ) + ( 2 ) ( 3 ) + ( 5 ) ( 9 ) ( 7 ) ( 10 ) + ( 9 ) ( 11 ) + ( 4 ) ( 2 ) ( 7 ) ( 1 ) + ( 9 ) ( 3 ) + ( 4 ) ( 9 ) ] C = \begin{bmatrix} (6)(10)+(2)(11)+(5)(2) & (6)(1)+(2)(3)+(5)(9) \\ (7)(10)+(9)(11)+(4)(2) & (7)(1)+(9)(3)+(4)(9) \end{bmatrix} C = [ ( 6 ) ( 10 ) + ( 2 ) ( 11 ) + ( 5 ) ( 2 ) ( 7 ) ( 10 ) + ( 9 ) ( 11 ) + ( 4 ) ( 2 ) ( 6 ) ( 1 ) + ( 2 ) ( 3 ) + ( 5 ) ( 9 ) ( 7 ) ( 1 ) + ( 9 ) ( 3 ) + ( 4 ) ( 9 ) ] C = [ 60 + 22 + 10 6 + 6 + 45 70 + 99 + 8 7 + 27 + 36 ] C = \begin{bmatrix} 60+22+10 & 6+6+45 \\ 70+99+8 & 7+27+36 \end{bmatrix} C = [ 60 + 22 + 10 70 + 99 + 8 6 + 6 + 45 7 + 27 + 36 ] C = [ 92 57 177 70 ] C = \begin{bmatrix} 92 & 57 \\ 177 & 70 \end{bmatrix} C = [ 92 177 57 70 ] C = [ 92 57 177 70 ] \boxed{C = \begin{bmatrix} 92 & 57 \\ 177 & 70 \end{bmatrix}} C = [ 92 177 57 70 ]
e) Find (i) C = A B C=AB C = A B D = B A D=BA D = B A
Given:
A = [ − 2 4 7 ] , B = [ 3 6 − 2 ] A = \begin{bmatrix} -2 \\ 4 \\ 7 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 6 & -2 \end{bmatrix} A = − 2 4 7 , B = [ 3 6 − 2 ]
(i) Find C = A B C=AB C = A B
Step 1: Calculate C = A B C=AB C = A B C = [ − 2 4 7 ] [ 3 6 − 2 ] C = \begin{bmatrix} -2 \\ 4 \\ 7 \end{bmatrix} \begin{bmatrix} 3 & 6 & -2 \end{bmatrix} C = − 2 4 7 [ 3 6 − 2 ] C = [ ( − 2 ) ( 3 ) ( − 2 ) ( 6 ) ( − 2 ) ( − 2 ) ( 4 ) ( 3 ) ( 4 ) ( 6 ) ( 4 ) ( − 2 ) ( 7 ) ( 3 ) ( 7 ) ( 6 ) ( 7 ) ( − 2 ) ] C = \begin{bmatrix} (-2)(3) & (-2)(6) & (-2)(-2) \\ (4)(3) & (4)(6) & (4)(-2) \\ (7)(3) & (7)(6) & (7)(-2) \end{bmatrix} C = ( − 2 ) ( 3 ) ( 4 ) ( 3 ) ( 7 ) ( 3 ) ( − 2 ) ( 6 ) ( 4 ) ( 6 ) ( 7 ) ( 6 ) ( − 2 ) ( − 2 ) ( 4 ) ( − 2 ) ( 7 ) ( − 2 ) C = [ − 6 − 12 4 12 24 − 8 21 42 − 14 ] C = \begin{bmatrix} -6 & -12 & 4 \\ 12 & 24 & -8 \\ 21 & 42 & -14 \end{bmatrix} C = − 6 12 21 − 12 24 42 4 − 8 − 14 C = [ − 6 − 12 4 12 24 − 8 21 42 − 14 ] \boxed{C = \begin{bmatrix} -6 & -12 & 4 \\ 12 & 24 & -8 \\ 21 & 42 & -14 \end{bmatrix}} C = − 6 12 21 − 12 24 42 4 − 8 − 14
(ii) Find D = B A D=BA D = B A
Step 1: Calculate D = B A D=BA D = B A D = [ 3 6 − 2 ] [ − 2 4 7 ] D = \begin{bmatrix} 3 & 6 & -2 \end{bmatrix} \begin{bmatrix} -2 \\ 4 \\ 7 \end{bmatrix} D = [ 3 6 − 2 ] − 2 4 7 D = [ ( 3 ) ( − 2 ) + ( 6 ) ( 4 ) + ( − 2 ) ( 7 ) ] D = \begin{bmatrix} (3)(-2)+(6)(4)+(-2)(7) \end{bmatrix} D = [ ( 3 ) ( − 2 ) + ( 6 ) ( 4 ) + ( − 2 ) ( 7 ) ] D = [ − 6 + 24 − 14 ] D = \begin{bmatrix} -6+24-14 \end{bmatrix} D = [ − 6 + 24 − 14 ] D = [ 4 ] D = \begin{bmatrix} 4 \end{bmatrix} D = [ 4 ] D = [ 4 ] \boxed{D = \begin{bmatrix} 4 \end{bmatrix}} D = [ 4 ]
