Equality of Matrices

Equality of Matrices — The Core Idea

Two matrices are equal only when they are identical in every respect — a strict condition, not mere similarity of shape. For two matrices $A$ and $B$ to be equal, two conditions must hold simultaneously.

Condition 1: Same Order

If $A$ is of order $m \times n$, then $B$ must also be of order $m \times n$. A $2 \times 3$ matrix can never equal a $3 \times 2$ matrix, whatever numbers are inside.

Condition 2: Corresponding Elements Equal

If $A = [a_{ij}]$ and $B = [b_{ij}]$, then for every row index $i$ and column index $j$,

If even a single pair of corresponding elements differs, the matrices are not equal.

When both conditions hold, we write $A = B$.

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Worked Illustrations

Illustration 1: Simple Comparison

are equal — same order ($2 \times 2$) and every corresponding element matches. However,

are not equal: both are $2 \times 2$, but $c_{11} = 3$ while $d_{11} = 2$.

Illustration 2: Finding Unknowns from Equality

If

$\begin{bmatrix} x & y \\ z & a \\ b & c \end{bmatrix} = \begin{bmatrix} -1.5 & 0 \\ 2 & 6 \\ 3 & 2 \end{bmatrix}$ …