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Calculate the matrix product C=AB for the given matrices A and B in each part.

Mathematics

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6 parts
a

C=[3613660190]C = \begin{bmatrix} 36 & 136 \\ 60 & 190 \end{bmatrix}

b

C=[267469297852]C = \begin{bmatrix} 26 & 74 & 69 \\ 29 & 78 & 52 \end{bmatrix}

c

C=[11794465162191387656]C = \begin{bmatrix} 117 & 94 & 46 \\ 51 & 62 & 19 \\ 138 & 76 & 56 \end{bmatrix}

d

C=[925717770]C = \begin{bmatrix} 92 & 57 \\ 177 & 70 \end{bmatrix}

e(i)

C=[−6−1241224−82142−14]C = \begin{bmatrix} -6 & -12 & 4 \\ 12 & 24 & -8 \\ 21 & 42 & -14 \end{bmatrix}

e(ii)

D=[4]D = \begin{bmatrix} 4 \end{bmatrix}

Step-by-step solution

6 steps

Here are the calculations for C=ABC=AB for each part.

Part a· 1 step

Find C=ABC=AB

Given:

A=[1214205],B=[3902]A = \begin{bmatrix} 12 & 14 \\ 20 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 9 \\ 0 & 2 \end{bmatrix}

  1. Calculate C=ABC=AB.

    C=[1214205][3902]C = \begin{bmatrix} 12 & 14 \\ 20 & 5 \end{bmatrix} \begin{bmatrix} 3 & 9 \\ 0 & 2 \end{bmatrix} 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=[36+0108+2860+0180+10]C = \begin{bmatrix} 36+0 & 108+28 \\ 60+0 & 180+10 \end{bmatrix} C=[3613660190]C = \begin{bmatrix} 36 & 136 \\ 60 & 190 \end{bmatrix}

    C=[3613660190]\boxed{C = \begin{bmatrix} 36 & 136 \\ 60 & 190 \end{bmatrix}}

Part b

C=[267469297852]C = \begin{bmatrix} 26 & 74 & 69 \\ 29 & 78 & 52 \end{bmatrix}

Find C=ABC=AB

Given:

A=[4791],B=[385267]A = \begin{bmatrix} 4 & 7 \\ 9 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 8 & 5 \\ 2 & 6 & 7 \end{bmatrix}

  1. Calculate C=ABC=AB.

    C=[4791][385267]C = \begin{bmatrix} 4 & 7 \\ 9 & 1 \end{bmatrix} \begin{bmatrix} 3 & 8 & 5 \\ 2 & 6 & 7 \end{bmatrix} 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=[12+1432+4220+4927+272+645+7]C = \begin{bmatrix} 12+14 & 32+42 & 20+49 \\ 27+2 & 72+6 & 45+7 \end{bmatrix} C=[267469297852]C = \begin{bmatrix} 26 & 74 & 69 \\ 29 & 78 & 52 \end{bmatrix}

    C=[267469297852]\boxed{C = \begin{bmatrix} 26 & 74 & 69 \\ 29 & 78 & 52 \end{bmatrix}}

Part c

C=[11794465162191387656]C = \begin{bmatrix} 117 & 94 & 46 \\ 51 & 62 & 19 \\ 138 & 76 & 56 \end{bmatrix}

Find C=ABC=AB

Given:

A=[71129106],B=[1245361]A = \begin{bmatrix} 7 & 11 \\ 2 & 9 \\ 10 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 12 & 4 & 5 \\ 3 & 6 & 1 \end{bmatrix}

  1. Calculate C=ABC=AB.

    C=[71129106][1245361]C = \begin{bmatrix} 7 & 11 \\ 2 & 9 \\ 10 & 6 \end{bmatrix} \begin{bmatrix} 12 & 4 & 5 \\ 3 & 6 & 1 \end{bmatrix} 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=[84+3328+6635+1124+278+5410+9120+1840+3650+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=[11794465162191387656]C = \begin{bmatrix} 117 & 94 & 46 \\ 51 & 62 & 19 \\ 138 & 76 & 56 \end{bmatrix}

    C=[11794465162191387656]\boxed{C = \begin{bmatrix} 117 & 94 & 46 \\ 51 & 62 & 19 \\ 138 & 76 & 56 \end{bmatrix}}

Part d

C=[925717770]C = \begin{bmatrix} 92 & 57 \\ 177 & 70 \end{bmatrix}

Find C=ABC=AB

Given:

A=[625794],B=[10111329]A = \begin{bmatrix} 6 & 2 & 5 \\ 7 & 9 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 10 & 1 \\ 11 & 3 \\ 2 & 9 \end{bmatrix}

  1. Calculate C=ABC=AB.

    C=[625794][10111329]C = \begin{bmatrix} 6 & 2 & 5 \\ 7 & 9 & 4 \end{bmatrix} \begin{bmatrix} 10 & 1 \\ 11 & 3 \\ 2 & 9 \end{bmatrix} 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=[60+22+106+6+4570+99+87+27+36]C = \begin{bmatrix} 60+22+10 & 6+6+45 \\ 70+99+8 & 7+27+36 \end{bmatrix} C=[925717770]C = \begin{bmatrix} 92 & 57 \\ 177 & 70 \end{bmatrix}

    C=[925717770]\boxed{C = \begin{bmatrix} 92 & 57 \\ 177 & 70 \end{bmatrix}}

Part e(i)

C=[−6−1241224−82142−14]C = \begin{bmatrix} -6 & -12 & 4 \\ 12 & 24 & -8 \\ 21 & 42 & -14 \end{bmatrix}

Find C=ABC=AB

Find (i) C=ABC=AB, and (ii) D=BAD=BA

Given:

A=[−247],B=[36−2]A = \begin{bmatrix} -2 \\ 4 \\ 7 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 6 & -2 \end{bmatrix}

  1. Calculate C=ABC=AB.

    C=[−247][36−2]C = \begin{bmatrix} -2 \\ 4 \\ 7 \end{bmatrix} \begin{bmatrix} 3 & 6 & -2 \end{bmatrix} 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=[−6−1241224−82142−14]C = \begin{bmatrix} -6 & -12 & 4 \\ 12 & 24 & -8 \\ 21 & 42 & -14 \end{bmatrix}

    C=[−6−1241224−82142−14]\boxed{C = \begin{bmatrix} -6 & -12 & 4 \\ 12 & 24 & -8 \\ 21 & 42 & -14 \end{bmatrix}}

Part e(ii)

D=[4]D = \begin{bmatrix} 4 \end{bmatrix}

Find D=BAD=BA

  1. Calculate D=BAD=BA.

    D=[36−2][−247]D = \begin{bmatrix} 3 & 6 & -2 \end{bmatrix} \begin{bmatrix} -2 \\ 4 \\ 7 \end{bmatrix} D=[(3)(−2)+(6)(4)+(−2)(7)]D = \begin{bmatrix} (3)(-2)+(6)(4)+(-2)(7) \end{bmatrix} D=[−6+24−14]D = \begin{bmatrix} -6+24-14 \end{bmatrix} D=[4]D = \begin{bmatrix} 4 \end{bmatrix}

    D=[4]\boxed{D = \begin{bmatrix} 4 \end{bmatrix}}

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