Equality of Matrices

The Condition for Equality

Two matrices cannot be compared entry by entry unless they are laid out the same way, so equality of matrices needs two conditions to both hold at once, not just one.

Definition. Two matrices $A=[a_{ij}]_{m\times n}$ and $B=[b_{ij}]_{p\times q}$ are said to be equal, written $A=B$, if and only if:

1. They have the same order, i.e. $m=p$ and $n=q$; and
2. Every pair of corresponding entries is equal, i.e. $a_{ij}=b_{ij}$ for every valid $i$ and $j$.

Both conditions are necessary. A $2\times3$ matrix can never equal a $3\times2$ matrix even if both happen to contain the same six numbers, because their orders differ; and two matrices of the same order are not equal unless every single corresponding entry matches, not just some.

Solving Equations from Matrix Equality

Because equality forces every corresponding pair of entries to match, a single matrix equation between two matrices containing unknowns is really a compact way of writing several ordinary equations at once -- one equation per entry. Setting up and solving this small system is the standard technique whenever unknowns appear inside a matrix equality, as in Example 3:

Matching entries in the $(1,1)$ position gives $x+y=6$; matching entries in the $(2,2)$ position gives $x-y=2$; the $(1,2)$ and $(2,1)$ positions are already equal ($2=2$ and $5=5$) and contribute no new information. The two equations $x+y=6$ and $x-y=2$ are then solved simultaneously in the ordinary way (by elimination or substitution) to find the values of $x$ and $y$. …