If A = [1221] and B =[2112],\begin{bmatrix}[r] 1 & 2 \ 2 & 1 \end{bmatrix} \text{ and B } = \begin{bmatrix}[r] 2 & 1 \ 1 & 2 \end{bmatrix},[1221] and B =[2112], find A(BA).
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BA =[2112][1221]=[2×1+1×22×2+1×11×1+2×21×2+2×1]=[2+24+11+42+2]=[4554]∴A(BA) =A×BA=[1221][4554]=[1×4+2×51×5+2×42×4+1×52×5+1×4]=[4+105+88+510+4]=[14131314]\text{BA } = \begin{bmatrix}[r] 2 & 1 \ 1 & 2 \end{bmatrix} \begin{bmatrix}[r] 1 & 2 \ 2 & 1 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 2 \times 1 + 1 \times 2 & 2 \times 2 + 1 \times 1 \ 1 \times 1 + 2 \times 2 & 1 \times 2 + 2 \times 1 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 2 + 2 & 4 + 1 \ 1 + 4 & 2 + 2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 4 & 5 \ 5 & 4 \end{bmatrix} \\[1.5em] \therefore \text{A(BA) } = \text{A} \times \text{BA} \\[1em] = \begin{bmatrix}[r] 1 & 2 \ 2 & 1 \end{bmatrix} \begin{bmatrix}[r] 4 & 5 \ 5 & 4 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 1 \times 4 + 2 \times 5 & 1 \times 5 + 2 \times 4 \ 2 \times 4 + 1 \times 5 & 2 \times 5 + 1 \times 4 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 4 + 10 & 5 + 8 \ 8 + 5 & 10 + 4 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 14 & 13 \ 13 & 14 \end{bmatrix}BA =[2112][1221]=[2×1+1×21×1+2×22×2+1×11×2+2×1]=[2+21+44+12+2]=[4554]∴A(BA) =A×BA=[1221][4554]=[1×4+2×52×4+1×51×5+2×42×5+1×4]=[4+108+55+810+4]=[14131314]
Hence, the matrix A(BA) = [14131314].\begin{bmatrix}[r] 14 & 13 \ 13 & 14 \end{bmatrix}.[14131314].
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If A = [2513], B =[1−1−32],\begin{bmatrix}[r] 2 & 5 \ 1 & 3 \end{bmatrix}, \text{ B } = \begin{bmatrix}[r] 1 & -1 \ -3 & 2 \end{bmatrix},[2153], B =[1−3−12], find AB and BA. Is AB = BA ?
If A = [3724], B =[0253] and C =[1−5−46],\begin{bmatrix}[r] 3 & 7 \ 2 & 4 \end{bmatrix}, \text{ B } = \begin{bmatrix}[r] 0 & 2 \ 5 & 3 \end{bmatrix} \text{ and C } = \begin{bmatrix}[r] 1 & -5 \ -4 & 6 \end{bmatrix},[3274], B =[0523] and C =[1−4−56], find AB - 5C.
Given the matrices :
A =[2142], B =[34−1−2] and C =[−310−2].\text { A } = \begin{bmatrix}[r] 2 & 1 \ 4 & 2 \end{bmatrix}, \text{ B } = \begin{bmatrix}[r] 3 & 4 \ -1 & -2 \end{bmatrix} \text{ and C } = \begin{bmatrix}[r] -3 & 1 \ 0 & -2 \end{bmatrix} . A =[2412], B =[3−14−2] and C =[−301−2].
Find the products of (i) ABC (ii) ACB and state whether they are equal.
Evaluate : [4 sin 30° 2 cos 60° sin 90° 2 cos 0°][4554].\begin{bmatrix} \text{4 sin 30° } & \text {2 cos 60°} \ \text{ sin 90° } & \text{ 2 cos 0°} \end{bmatrix} \begin{bmatrix} 4 & 5 \ 5 & 4 \end{bmatrix}.[4 sin 30° sin 90° 2 cos 60° 2 cos 0°][4554].
