Q.If $A = \begin{bmatrix} 2 & -3 \\ 5 & -4 \\ -6 & 1 \end{bmatrix}$, $B = \begin{bmatrix} -1 & 2 \\ 2 & 2 \\ 0 & 3 \end{bmatrix}$ and $C = \begin{bmatrix} 4 & 3 \\ -1 & 4 \\ -2 & 1 \end{bmatrix}$, show that $A+B = B+A$.

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 $X$ exactly as you would an unknown number, isolating it algebraically, then compute the resulting combination of known matrices entry-by-entry. These "sum-and-scale" operations are the foundation on top of which matrix multiplication (a genuinely different, non-commutative operation) is built.