The matrix multiplication introduction page demonstrated how to approach multiplying same size matrices together.
We can also perform the action of multiplying matrices of different sizes together sometimes, when certain conditions are met.
Multiplying Matrices of Different Sizes
When you can do it
A matrix size can be denoted by:
( ROWS × COLUMNS ).
So if we have 2 matrices ( m × n ) and ( p × q ),
and if we want to multiply them together, ( m × n ) × ( p × q ).
The first step is to check that n and p are equal.
Which is checking that the number of columns in the first matrix equals the amount of rows in the second.
Then if this is satisfied, the matrices can be multiplied, and the resulting matrix from the multiplication will be of size ( m × q ).
This will probably look a bit clearer with an example involving numbers.
Example
(1.1)
\begin{bmatrix} 2 & 4 & 3 \\ 1 & 2 & 1 \end{bmatrix} × \begin{bmatrix} 2 & 4 \\ 3 & 1 \\ 5 & 4 \end{bmatrix}
Solution
This is a ( 2 × 3 ) matrix multiplying a ( 3 × 2 ) matrix.
So the first matrix has the same number of columns as the second matrix has rows, which is what we need.
We can also see that the resulting matrix from multiplication should be a ( 2 × 2 ) matrix.
\begin{bmatrix} 2 & 4 & 3 \\ 1 & 2 & 1 \end{bmatrix} × \begin{bmatrix} 2 & 4 \\ 3 & 1 \\ 5 & 4 \end{bmatrix}
= \begin{bmatrix} 2.2+4.3+3.5 & 2.4+4.1+3.4 \\ 1.2+2.3+1.5 & 1.4+2.1+1.4 \end{bmatrix}
= \begin{bmatrix} 4+12+15 & 8+4+12 \\ 2+6+5 & 4+2+4 \end{bmatrix} = \begin{bmatrix} 31 & 24 \\ 13 & 10 \end{bmatrix}
(1.2)
\begin{bmatrix} 1 & 4 \\ {\text{-}}2 & 3 \\ 0 & 1\end{bmatrix} × \begin{bmatrix} {\text{-}}3 & 7 & 1 \\ 2 & 0 & {\text{-}}2\end{bmatrix}
Solution
This time we have a ( 3 × 2 ) matrix multiplying a ( 2 × 3 ) matrix.
The resulting matrix from multiplication will be a ( 3 × 3 ) matrix.
\begin{bmatrix} 1 & 4 \\ {\text{-}}2 & 3 \\ 0 & 1\end{bmatrix} × \begin{bmatrix} {\text{-}}3 & 7 & 1 \\ 2 & 0 & {\text{-}}2\end{bmatrix}
= \begin{bmatrix} 1.{\text{-}}3+4.2 & 1.7+4.0 & 1.1+4{\text{-}}2 \\ {\text{-}}2.{\text{-}}3+3.2 & {\text{-}}2.7+3.0 & {\text{-}}2.1+3.{\text{-}}2 \\ 0.{\text{-}}3+1.2 & 0.7+1.0 & 0.1+1.{\text{-}}2 \end{bmatrix}
= \begin{bmatrix} {\text{-}}3+8 & 7+0 & 1{\text{--}}8 \\ 6+6 & {\text{-}}14+0 & {\text{-}}2{\text{--}}6 \\ 0+2 & 0+0 & 0{\text{--}}2 \end{bmatrix} = \begin{bmatrix} 5 & 7 & {\text{-}}7 \\ 12 & {\text{-}}14 & {\text{-}}8 \\ 2 & 0 & {\text{-}}2 \end{bmatrix}