If $X = \begin{bmatrix} 3 & 1 & -1 \\ 5 & -2 & -3 \end{bmatrix}$ and $Y = \begin{bmatrix} 2 & 1 & -1 \\ 7 & 2 & 4 \end{bmatrix}$ then find a matrix $Z$ such that $X + Y + Z$ is a zero matrix.

$\begin{bmatrix} -5 & -2 & 2 \\ -12 & 0 & -1 \end{bmatrix}$

The correct answer is Option (1) → $\begin{bmatrix} -5 & -2 & 2 \\ -12 & 0 & -1 \end{bmatrix}$ ##

We have, $X = \begin{bmatrix} 3 & 1 & -1 \\ 5 & -2 & -3 \end{bmatrix}_{2 \times 3}$ and $Y = \begin{bmatrix} 2 & 1 & -1 \\ 7 & 2 & 4 \end{bmatrix}_{2 \times 3}$

$X + Y = \begin{bmatrix} 3 + 2 & 1 + 1 & -1 - 1 \\ 5 + 7 & -2 + 2 & -3 + 4 \end{bmatrix} = \begin{bmatrix} 5 & 2 & -2 \\ 12 & 0 & 1 \end{bmatrix}$

Also, $X + Y + Z = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$

We see that $Z$ is the additive inverse of $(X + Y)$ or negative of $(X + Y)$.

$∴Z = \begin{bmatrix} -5 & -2 & 2 \\ -12 & 0 & -1 \end{bmatrix} \quad [∵Z = -(X + Y)]$