If A = [1−22−1]and B =[32−21],\begin{bmatrix}[r] 1 & -2 \ 2 & -1 \end{bmatrix} \text{and B } = \begin{bmatrix}[r] 3 & 2 \ -2 & 1 \end{bmatrix},[12−2−1]and B =[3−221], find 2B - A2.
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2B=2[32−21]=[64−42]A2=[1−22−1][1−22−1]=[1×1+(−2)×21×(−2)+(−2)×(−1)2×1+(−1)×22×(−2)+(−1)×(−1)]=[1−4−2+22−2−4+1]=[−300−3]∴2B−A2=[64−42]−[−300−3]=[6−(−3)4−0−4−02−(−3)]=[94−45]2B = 2\begin{bmatrix}[r] 3 & 2 \ -2 & 1 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 6 & 4 \ -4 & 2 \end{bmatrix} \\[1em] A^2 = \begin{bmatrix}[r] 1 & -2 \ 2 & -1 \end{bmatrix} \begin{bmatrix}[r] 1 & -2 \ 2 & -1 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 1 \times 1 + (-2) \times 2 & 1 \times (-2) + (-2) \times (-1) \ 2 \times 1 + (-1) \times 2 & 2 \times (-2) + (-1)\times (-1) \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 1 - 4 & -2 + 2 \ 2 - 2 & -4 + 1 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -3 & 0 \ 0 & -3 \end{bmatrix} \\[1em] \therefore 2B - A^2 = \begin{bmatrix}[r] 6 & 4 \ -4 & 2 \end{bmatrix} - \begin{bmatrix}[r] -3 & 0 \ 0 & -3 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 6 - (-3) & 4 - 0 \ -4 - 0 & 2 - (-3) \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 9 & 4 \ -4 & 5 \end{bmatrix}2B=2[3−221]=[6−442]A2=[12−2−1][12−2−1]=[1×1+(−2)×22×1+(−1)×21×(−2)+(−2)×(−1)2×(−2)+(−1)×(−1)]=[1−42−2−2+2−4+1]=[−300−3]∴2B−A2=[6−442]−[−300−3]=[6−(−3)−4−04−02−(−3)]=[9−445]
Hence, the matrix 2B - A2 = [94−45].\begin{bmatrix}[r] 9 & 4 \ -4 & 5 \end{bmatrix}.[9−445].
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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].
If A = [−1324] and B =[2−3−4−6],\begin{bmatrix}[r] -1 & 3 \ 2 & 4 \end{bmatrix} \text{ and B } = \begin{bmatrix}[r] 2 & -3 \ -4 & -6 \end{bmatrix},[−1234] and B =[2−4−3−6], find the matrix AB + BA.
If A = [1234], B =[2142] and C =[5174],\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \text{ B } = \begin{bmatrix} 2 & 1 \ 4 & 2 \end{bmatrix} \text{ and C } = \begin{bmatrix} 5 & 1 \ 7 & 4 \end{bmatrix},[1324], B =[2412] and C =[5714], compute
(i) A(B + C)
(ii) (B + C)A
If A = [1223], B =[2132] and C =[1331],\begin{bmatrix} 1 & 2 \ 2 & 3 \end{bmatrix}, \text{ B } = \begin{bmatrix} 2 & 1 \ 3 & 2 \end{bmatrix} \text{ and C } = \begin{bmatrix} 1 & 3 \ 3 & 1 \end{bmatrix},[1223], B =[2312] and C =[1331], find the matrix C(B - A).
