Matrices

Check the inner dimensions
If A has order \(m\times n\) and B has order \(n\times p\), then AB exists and has order \(m\times p\). The number of columns of A must equal the number of rows of B. Check this before calculating.
Multiply a row by a column
The entry in row i, column j of AB is found by multiplying corresponding entries of row i of A and column j of B, then adding. For example, \(\begin{pmatrix}2&3\end{pmatrix}\begin{pmatrix}4\\ 5\end{pmatrix}=\begin{pmatrix}2(4)+3(5)\end{pmatrix}=\begin{pmatrix}23\end{pmatrix}\).
Matrix multiplication is not entrywise multiplication. In general \(AB\ne BA\); sometimes only one of the two products exists. For a square matrix, \(A^2\) means \(AA\), not squaring each entry.
When an entry is unknown, write the appropriate row-column equation. Solve that ordinary equation and check it in the product.

If A has order \(m\times n\) and B has order \(n\times p\), then AB exists and has order \(m\times p\). The number of columns of A must equal the number of rows of B. Check this before calculating.

The entry in row i, column j of AB is found by multiplying corresponding entries of row i of A and column j of B, then adding. For example, \(\begin{pmatrix}2&3\end{pmatrix}\begin{pmatrix}4\\ 5\end{pmatrix}=\begin{pmatrix}2(4)+3(5)\end{pmatrix}=\begin{pmatrix}23\end{pmatrix}\).

Matrix multiplication is not entrywise multiplication. In general \(AB\ne BA\); sometimes only one of the two products exists. For a square matrix, \(A^2\) means \(AA\), not squaring each entry.

When an entry is unknown, write the appropriate row-column equation. Solve that ordinary equation and check it in the product.

Matrix products and their order

Matrix products and their order