Given A = $\begin{bmatrix} 2 & -3 \end{bmatrix}, B = \begin{bmatrix} 0 & 2 \end{bmatrix} \text{ and C} = \begin{bmatrix} -1 & 4 \end{bmatrix};$ find the matrix X in each of the following :

(i) X + B = C - A

(ii) A - X = B + C

(i) Given,

⇒ X + B = C - A

⇒ X = C - A - B

Substituting values of A, B and C in above equation we get,

$\Rightarrow X = \begin{bmatrix} -1 & 4 \end{bmatrix} - \begin{bmatrix} 2 & -3 \end{bmatrix} - \begin{bmatrix} 0 & 2 \end{bmatrix} \\[1em] = \begin{bmatrix} -1 - 2 - 0 & 4 - (-3) - 2 \end{bmatrix} \\[1em] = \begin{bmatrix} -3 & 5 \end{bmatrix}.$

Hence, X = $\begin{bmatrix} -3 & 5 \end{bmatrix}.$

(ii) Given,

⇒ A - X = B + C

⇒ X = A - (B + C)

Substituting values of A, B and C in above equation we get,

$\Rightarrow X = \begin{bmatrix} 2 & -3 \end{bmatrix} - \Big(\begin{bmatrix} 0 & 2 \end{bmatrix} + \begin{bmatrix} -1 & 4 \end{bmatrix}\Big) \\[1em] = \begin{bmatrix} 2 & -3 \end{bmatrix} - \Big(\begin{bmatrix} 0 + (-1) & 2 + 4 \end{bmatrix}\Big) \\[1em] = \begin{bmatrix} 2 & -3 \end{bmatrix} - \begin{bmatrix} -1 & 6 \end{bmatrix} \\[1em] = \begin{bmatrix} 2 - (-1) & -3 - 6 \end{bmatrix} \\[1em] = \begin{bmatrix} 3 & -9 \end{bmatrix}.$

Hence, X = $\begin{bmatrix} 3 & -9 \end{bmatrix}$.