Given : M = [5−3−24]\begin{bmatrix} 5 & -3 \ -2 & 4 \end{bmatrix}[5−2−34], find its transpose matrix Mt. If possible, find :
(i) M + Mt
(ii) Mt - M
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M = [5−3−24]\begin{bmatrix} 5 & -3 \ -2 & 4 \end{bmatrix}[5−2−34]
Mt = [5−2−34]\begin{bmatrix} 5 & -2 \ -3 & 4 \end{bmatrix}[5−3−24]
(i)
M+Mt=[5−3−24]+[5−2−34]=[5+5−3+(−2)−2+(−3)4+4]=[10−5−58].M + M^t = \begin{bmatrix} 5 & -3 \ -2 & 4 \end{bmatrix} + \begin{bmatrix} 5 & -2 \ -3 & 4 \end{bmatrix} \\[1em] = \begin{bmatrix} 5 + 5 & -3 + (-2) \ -2 + (-3) & 4 + 4 \end{bmatrix} \\[1em] = \begin{bmatrix} 10 & -5 \ -5 & 8 \end{bmatrix}.M+Mt=[5−2−34]+[5−3−24]=[5+5−2+(−3)−3+(−2)4+4]=[10−5−58].
Hence, M+Mt=[10−5−58].M + M^t = \begin{bmatrix} 10 & -5 \ -5 & 8 \end{bmatrix}.M+Mt=[10−5−58].
(ii)
Mt−M=[5−2−34]−[5−3−24]=[5−5−2−(−3)−3−(−2)4−4]=[01−10].M^t - M = \begin{bmatrix} 5 & -2 \ -3 & 4 \end{bmatrix} - \begin{bmatrix} 5 & -3 \ -2 & 4 \end{bmatrix} \\[1em] = \begin{bmatrix} 5 - 5 & -2 - (-3) \ -3 - (-2) & 4 - 4 \end{bmatrix} \\[1em] = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}.Mt−M=[5−3−24]−[5−2−34]=[5−5−3−(−2)−2−(−3)4−4]=[0−110].
Hence, Mt - M = [01−10].\begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix}.[0−110].
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Find, x and y from the following equations :
[52−1y−1]−[1x−12−3]=[47−32]\begin{bmatrix} 5 & 2 \ -1 & y - 1 \end{bmatrix} - \begin{bmatrix} 1 & x - 1 \ 2 & -3 \end{bmatrix} = \begin{bmatrix} 4 & 7 \ -3 & 2 \end{bmatrix}[5−12y−1]−[12x−1−3]=[4−372]
[−8x]+[y−2]=[−32]\begin{bmatrix} -8 & x \end{bmatrix} + \begin{bmatrix} y & -2 \end{bmatrix} = \begin{bmatrix} -3 & 2 \end{bmatrix}[−8x]+[y−2]=[−32]
Given A = [2−3],B=[02] and C=[−14];\begin{bmatrix} 2 & -3 \end{bmatrix}, B = \begin{bmatrix} 0 & 2 \end{bmatrix} \text{ and C} = \begin{bmatrix} -1 & 4 \end{bmatrix};[2−3],B=[02] and C=[−14]; find the matrix X in each of the following :
(i) X + B = C - A
(ii) A - X = B + C
Given A = [−102−4] and B=[3−3−20]\begin{bmatrix} -1 & 0 \ 2 & -4 \end{bmatrix} \text{ and } B = \begin{bmatrix} 3 & -3 \ -2 & 0 \end{bmatrix}[−120−4] and B=[3−2−30]; find the matrix X in each of the following :
(i) A + X = B
(ii) A - X = B
(iii) X - B = A
