How Do You Add, Subtract, and Multiply Matrices?

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For addition and subtraction, the order does not change. In symbols, if $A=(a_{ij})$ and $B=(b_{ij})$ have the same order, then $(A+B)_{ij}=a_{ij}+b_{ij}$ and $(A-B)_{ij}=a_{ij}-b_{ij}$. If their orders differ, neither operation is defined.

Let

Then

Using the same matrices, reversing the order gives $BA=\begin{pmatrix}16&3\\-7&-13\end{pmatrix}$, which differs from $AB$. This demonstrates that order matters even when both products exist. Before using a GDC, predict the output order from the outer dimensions. Afterwards, check that every entry uses one complete row-column pairing; this catches omitted terms and sign errors, especially with negative entries.

A common misconception is that matrix multiplication is entry-by-entry. It is not. Matrix multiplication is also generally non-commutative, so $AB\neq BA$, and one product may exist when the other does not.

Exam technique: State the dimensions before multiplying and show at least one row-by-column calculation. A GDC can perform matrix operations, but calculator output alone may not demonstrate the required method. Keep the matrices in the stated order and check the dimensions of the result.