1. Multiplying Matrices Together
2. Matrix Multiplication Steps
3. Multiplying by the Identity Matrix
4. Matrix Multiplication Properties
Matrix multiplication becomes a bit more involved when we look at multiplying matrices examples where matrices are multiplied together.
But it’s usually quite straightforward when we just want to multiply a matrix by a single number.
2 × \begin{bmatrix} 4 & 1 & 2 \\ 3 & 5 & 1 \end{bmatrix} = \begin{bmatrix} 2\times4 & 2\times1 & 2\times2 \\ 2\times3 & 2\times5 & 2\times1 \end{bmatrix} = \begin{bmatrix} 8 & 2 & 4 \\ 6 & 10 & 2 \end{bmatrix}
We just multiply each element of the matrix by the number.
Things are a little bit more complex when we want to multiply matrices together however.
Multiplying Matrices Examples
Same Size Matrix
We can look at the case of multiplying a 2 x 2 matrix by another 2 x 2 matrix.
\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} × \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}= \begin{bmatrix} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22} \end{bmatrix}
Now to see an example using numbers, with . representing multiplication, this looks like:
\begin{bmatrix} 1 & 3 \\ {\text{-}}4 & 1 \end{bmatrix} × \begin{bmatrix} 5 & 2 \\ 1 & 3 \end{bmatrix}
= \begin{bmatrix} 1.5+3.1 & 1.2+3.3 \\ {\text{-}}4.5+1.1 & {\text{-}}4.2+1.3 \end{bmatrix} = \begin{bmatrix} 8 & 11 \\ {\text{-}}19 & {\text{-}}5 \end{bmatrix}
The result of multiplying two 2 x 2 matrices together was another 2 x 2 matrix.
But generally the size of the new matrix resulting from the multiplication, depends on how many rows and columns are in the matrices being multiplied.
Along with the fact that matrices can only be multiplied together if one matrix has the same number of columns as the other does rows.
As such square matrices of the same size can always be multiplied together.
Matrix Order of Multiplication
It’s important to pay attention to order when learning matrix multiplication.
As if you have two matrices A and B, generally A×B ≠ B×A.
We can look at the multiplication of two different 2 x 2 matrices together.
=> \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} × \begin{bmatrix} 3 & 0 \\ 1 & 4 \end{bmatrix} = \begin{bmatrix} 1.3 + 2.1 & 1.0 + 2.4 \\ 0.3 + 1.1 & 0.0 + 1.4 \end{bmatrix} = \begin{bmatrix} 5 & 8 \\ 1 & 4 \end{bmatrix}
=> \begin{bmatrix} 3 & 0 \\ 1 & 4 \end{bmatrix} × \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 3.1 + 0.0 & 3.2 + 0.1 \\ 1.1 + 4.0 & 1.2 + 4.1 \end{bmatrix} = \begin{bmatrix} 3 & 6 \\ 1 & 6 \end{bmatrix}
( When practicing how to do matrix multiplication it’s handy to be able to quickly check your answers.
A handy matrix multiplication calculator is available to use at the matrix reshish website. )
Identity Matrix Multiplication
A specific case to consider when learning how to do matrix multiplication, is that multiplication of a matrix with the identity matrix, regardless of order, results in the same matrix.
\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} × \begin{bmatrix} 4 & 3 \\ 1 & 9 \end{bmatrix} = \begin{bmatrix} 4+0 & 3+0 \\ 0+1 & 0+9 \end{bmatrix}
= \begin{bmatrix} 4 & 3 \\ 1 & 9 \end{bmatrix}
\begin{bmatrix} 4 & 3 \\ 1 & 9 \end{bmatrix} × \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 4+0 & 0+3 \\ 1+0 & 0+9 \end{bmatrix}
= \begin{bmatrix} 4 & 3 \\ 1 & 9 \end{bmatrix}
This example was just with a simple ( 2 × 2 ) matrix, but the result will be the same with larger matrices also.
Properties of Matrix Multiplication
To round off this how to do matrix multiplication page we’ll display a short list of properties of matrix multiplication as a summary.
1) A × ( BC ) = AB × C ( associative property )
2) cA × B = A × cB ( where c is a constant )
3) AI = A , IA = A ( where I is the identity matrix )