Equality of Matrices

Two matrices $A=[a_{ij}]$ and $B=[b_{ij}]$ are equal, written $A=B$, exactly when:

1. both matrices have the same order (equal number of rows and equal number of columns), and
2. every corresponding entry is equal: $a_{ij}=b_{ij}$ for all $i$ and $j$.

If either condition fails — the orders don't match, or at least one corresponding pair of entries disagrees — the matrices are unequal.

This definition is what turns a matrix equation into ordinary algebra: given $\begin{pmatrix} x-2.5 & y \\ 1/5 & v\end{pmatrix} = \begin{pmatrix} 1 & -1 \\ 3/2 & 2 \end{pmatrix}$ (say), matching entry-by-entry immediately yields the four scalar equations $x-2.5=1$, $y=-1$, $\tfrac15=\tfrac32$ (impossible — so such a system may over-determine or be consistent depending on the numbers actually given), and $v=2$. In a genuine consistent problem, com …