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Mathematics

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}; find the matrix X in each of the following :

(i) A + X = B

(ii) A - X = B

(iii) X - B = A

Matrices

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Answer

(i) Given,

⇒ A + X = B

⇒ X = B - A

⇒X=[3−3−20]−[−102−4]=[3−(−1)−3−0−2−20−(−4)]=[4−3−44].\Rightarrow X = \begin{bmatrix} 3 & -3 \ -2 & 0 \end{bmatrix} - \begin{bmatrix} -1 & 0 \ 2 & -4 \end{bmatrix} \\[1em] = \begin{bmatrix} 3 - (-1) & -3 - 0 \ -2 - 2 & 0 - (-4) \end{bmatrix} \\[1em] = \begin{bmatrix} 4 & -3 \ -4 & 4 \end{bmatrix}.

Hence, X = [4−3−44].\begin{bmatrix} 4 & -3 \ -4 & 4 \end{bmatrix}.

(ii) Given,

⇒ A - X = B

⇒ X = A - B

⇒X=[−102−4]−[3−3−20]=[−1−30−(−3)2−(−2)−4−0]=[−434−4].\Rightarrow X = \begin{bmatrix} -1 & 0 \ 2 & -4 \end{bmatrix} - \begin{bmatrix} 3 & -3 \ -2 & 0 \end{bmatrix} \\[1em] = \begin{bmatrix} -1 - 3 & 0 - (-3) \ 2 - (-2) & -4 - 0 \end{bmatrix} \\[1em] = \begin{bmatrix} -4 & 3 \ 4 & -4 \end{bmatrix}.

Hence, X = [−434−4].\begin{bmatrix} -4 & 3 \ 4 & -4 \end{bmatrix}.

(iii) Given,

⇒ X - B = A

⇒ X = A + B

⇒X=[−102−4]+[3−3−20]=[−1+30+(−3)2+(−2)−4+0]=[2−30−4].\Rightarrow X = \begin{bmatrix} -1 & 0 \ 2 & -4 \end{bmatrix} + \begin{bmatrix} 3 & -3 \ -2 & 0 \end{bmatrix} \\[1em] = \begin{bmatrix} -1 + 3 & 0 + (-3) \ 2 + (-2) & -4 + 0 \end{bmatrix} \\[1em] = \begin{bmatrix} 2 & -3 \ 0 & -4 \end{bmatrix}.

Hence, X = [2−30−4].\begin{bmatrix} 2 & -3 \ 0 & -4 \end{bmatrix}.

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