Q.If $A = \begin{bmatrix} 1 & 2 & -3 \\ -3 & 7 & -8 \\ 0 & -6 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 9 & -1 & 2 \\ -4 & 2 & 5 \\ 4 & 0 & -3 \end{bmatrix}$, then find the matrix $C$ such that $A+B+C$ is a zero matrix.

Matrices of the same order can be checked for equality ($A=B$ iff every corresponding entry matches), added ($A+B=[a_{ij}+b_{ij}]$, entry-by-entry, undefined for mismatched orders), subtracted ($A-B=A+(-B)$), and scaled by a number ($kA=[k\,a_{ij}]$, every entry multiplied by $k$). These operations obey familiar-looking laws — addition is commutative ($A+B=B+A$) and associative ($(A+B)+C=A+(B+C)$), the zero matrix is the additive identity, $-A$ is the additive inverse, and scalar multiplication distributes over both matrix sums and scalar sums. The most common use is solving a linear matrix equation for an unknown matrix $X$ (e.g. $3A-4B+5X=C$): treat $X$ exac …