Question

Is matrix multiplication commutative?

Answer 1

Step 1: Assigning two matrices for multiplication

The commutative property of multiplication is defined as $AB=BA.$

Let there be two matrices $A$ and $B$ such that $A=\left[\begin{array}{cc}1& 4\\ 6& 7\end{array}\right]andB=\left[\begin{array}{cc}3& 4\\ 5& 7\end{array}\right]$

Now, multiplication of $A$ and $B$ is possible only if the number of columns of $A$ is equal to the number of rows of $B$. In the above case, this condition is satisfied.

Step 2: Calculating $AB$

$$\begin{gathered} AB=\left[\begin{array}{cc}1& 4\\ 6& 7\end{array}\right]\times \left[\begin{array}{cc}3& 4\\ 5& 7\end{array}\right] \\ =\left[\begin{array}{cc}3+20& 4+28\\ 18+35& 24+49\end{array}\right] \\ =\left[\begin{array}{cc}23& 32\\ 53& 73\end{array}\right] \end{gathered}$$

Step 3: Calculating $BA:$

$$\begin{gathered} BA=\left[\begin{array}{cc}3& 4\\ 5& 7\end{array}\right]\times \left[\begin{array}{cc}1& 4\\ 6& 7\end{array}\right] \\ =\left[\begin{array}{cc}3+24& 12+28\\ 5+42& 20+49\end{array}\right] \\ =\left[\begin{array}{cc}27& 40\\ 47& 69\end{array}\right] \\ \ne AB \end{gathered}$$

Therefore, we can say that matrix multiplication is not commutative in general.

Step 4: Finding with one of the matrices as an identity matrix

$$\begin{gathered} AB=\left[\begin{array}{cc}1& 4\\ 6& 7\end{array}\right]\times \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ =\left[\begin{array}{cc}1+0& 0+4\\ 6+0& 0+7\end{array}\right] \\ =\left[\begin{array}{cc}1& 4\\ 6& 7\end{array}\right] \end{gathered}$$

$$\begin{gathered} BA=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\times \left[\begin{array}{cc}1& 4\\ 6& 7\end{array}\right] \\ =\left[\begin{array}{cc}1+0& 4+0\\ 0+6& 0+7\end{array}\right] \\ =\left[\begin{array}{cc}1& 4\\ 6& 7\end{array}\right] \\ =AB \end{gathered}$$

Hence, it is clearly understood that for a matrix multiplication to be commutative, one of the matrices should be either an identity matrix or zero matrix.

In general, matrix multiplication is not commutative