Given A = [1423] and B=[−4−1−3−2]\begin{bmatrix}[r] 1 & 4 \ 2 & 3 \end{bmatrix} \text{ and } B = \begin{bmatrix}[r] -4 & -1 \ -3 & -2 \end{bmatrix}[1243] and B=[−4−3−1−2]
(i) find the matrix 2A + B
(ii) find a matrix C such that :
C + B = [0000]\begin{bmatrix}[r] 0 & 0 \ 0 & 0 \end{bmatrix}[0000]
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(i)
2A+B=2[1423]+[−4−1−3−2]=[2846]+[−4−1−3−2]=[2+(−4)8+(−1)4+(−3)6+(−2)]=[−2714].2A + B = 2\begin{bmatrix}[r] 1 & 4 \ 2 & 3 \end{bmatrix} + \begin{bmatrix}[r] -4 & -1 \ -3 & -2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 2 & 8 \ 4 & 6 \end{bmatrix} + \begin{bmatrix}[r] -4 & -1 \ -3 & -2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 2 + (-4) & 8 + (-1) \ 4 + (-3) & 6 + (-2) \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -2 & 7 \ 1 & 4 \end{bmatrix}.2A+B=2[1243]+[−4−3−1−2]=[2486]+[−4−3−1−2]=[2+(−4)4+(−3)8+(−1)6+(−2)]=[−2174].
Hence, 2A + B = [−2714]\begin{bmatrix}[r] -2 & 7 \ 1 & 4 \end{bmatrix}[−2174].
(ii) Given,
⇒C+B=[0000]⇒C=[0000]−B⇒C=[0000]−[−4−1−3−2]⇒C=[0−(−4)0−(−1)0−(−3)0−(−2)]⇒C=[4132].\Rightarrow C + B = \begin{bmatrix}[r] 0 & 0 \ 0 & 0 \end{bmatrix} \\[1em] \Rightarrow C = \begin{bmatrix}[r] 0 & 0 \ 0 & 0 \end{bmatrix} - B \\[1em] \Rightarrow C = \begin{bmatrix}[r] 0 & 0 \ 0 & 0 \end{bmatrix} - \begin{bmatrix}[r] -4 & -1 \ -3 & -2 \end{bmatrix} \\[1em] \Rightarrow C = \begin{bmatrix}[r] 0 - (-4) & 0 - (-1) \ 0 - (-3) & 0 - (-2) \end{bmatrix} \\[1em] \Rightarrow C = \begin{bmatrix}[r] 4 & 1 \ 3 & 2 \end{bmatrix}.⇒C+B=[0000]⇒C=[0000]−B⇒C=[0000]−[−4−3−1−2]⇒C=[0−(−4)0−(−3)0−(−1)0−(−2)]⇒C=[4312].
Hence, C = [4132]\begin{bmatrix}[r] 4 & 1 \ 3 & 2 \end{bmatrix}[4312].
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(ii) A + 2C - B
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If 2[3x01]+3[13y2]=[z−7158]2\begin{bmatrix}[r] 3 & x \ 0 & 1 \end{bmatrix} + 3\begin{bmatrix}[r] 1 & 3 \ y & 2 \end{bmatrix} = \begin{bmatrix}[r] z & -7 \ 15 & 8 \end{bmatrix}2[30x1]+3[1y32]=[z15−78]; find the values of x, y and z.
Given A = [−360−9]\begin{bmatrix}[r] -3 & 6 \ 0 & -9 \end{bmatrix}[−306−9] and At is its transpose matrix. Find :
(i) 2A + 3At
(ii) 2At - 3A
(iii) 12A−13At\dfrac{1}{2}A - \dfrac{1}{3}A^t21A−31At
(iv) At−13AA^t - \dfrac{1}{3}AAt−31A
