Question

If \(A=\begin{bmatrix}-2 & 1&-4 \\-2 & 4&-1 \\ 1 &-1 &-4 \end{bmatrix} \text{ and } B=\begin{bmatrix}-2 & 4&2 \\-4 & -1&1 \\ 4 &1 &1 \end{bmatrix}\)
then AB=?
BA=?
True or false : AB=BA for any two square matrices A and B of the same size.

Answer 1

Step 1
The given matrices are,
\(A=\begin{bmatrix}-2 & 1&-4 \\-2 & 4&-1 \\ 1 &-1 &-4 \end{bmatrix} \text{ and } B=\begin{bmatrix}-2 & 4&2 \\-4 & -1&1 \\ 4 &1 &1 \end{bmatrix}\)
Step 2
To find the value of AB, BA and telling whether AB=BA is true or not for any two square matrices of the same size.
Step 3
Taking the product of matrices, and evaluating AB,
\(AB=\begin{bmatrix}-2 & 1&-4 \\-2 & 4&-1 \\ 1 &-1 &-4 \end{bmatrix}\begin{bmatrix}-2 & 4&2 \\-4 & -1&1 \\ 4 &1 &1 \end{bmatrix}\)
\(=\begin{bmatrix}(-2\cdot-2)+(1\cdot-4)+(-4\cdot4) & (-2\cdot4)+(1\cdot-1)+(-4\cdot1)&(-2\cdot2)+(1\cdot1)+(-4\cdot1) \\(-2\cdot-2)+(4\cdot-4)+(-1\cdot4) & (-2\cdot4)+(4\cdot-1)+(-1\cdot1)&(-2\cdot2)+(4\cdot1)+(-1\cdot1) \\ (1\cdot-2)+(-1\cdot-4)+(-4\cdot4) &(1\cdot4)+(-1\cdot-1)+(-4\cdot1) &(1\cdot2)+(-1\cdot1)+(-4\cdot1) \end{bmatrix}\)
\(=\begin{bmatrix}4-4+-16 & -8-1-4&-4+1-4 \\ 4-16-4 & -8-4-1&-4+4-1\\-2+4+-16&4+1-4&2-1-4 \end{bmatrix}\)
\(=\begin{bmatrix}-16 & -13&-7 \\ -16 & -13&-1\\-14&1&-3 \end{bmatrix} \dots (1)\)
Step 4
Similarly, evaluating BA
\(BA=\begin{bmatrix}-2 & 4&2 \\-4 & -1&1 \\ 4 &1 &1 \end{bmatrix}\begin{bmatrix}-2 & 1&-4 \\-2 & 4&-1 \\ 1 &-1 &-4 \end{bmatrix}\)
\(=\begin{bmatrix}(-2\cdot-2)+(4\cdot-2)+(2\cdot1) & (-2\cdot1)+(4\cdot4)+(2\cdot-1)&(-2\cdot-4)+(4\cdot-1)+(2\cdot-4) \\(-4\cdot-2)+(-1\cdot-2)+(1\cdot1) & (-4\cdot1)+(-1\cdot4)+(1\cdot-1)&(-4\cdot-4)+(-1\cdot-1)+(1\cdot-4) \\ (4\cdot-2)+(1\cdot-2)+(1\cdot1) &(4\cdot1)+(1\cdot4)+(1\cdot-1) &(4\cdot-4)+(1\cdot-1)+(1\cdot-4) \end{bmatrix}\)
\(=\begin{bmatrix}4-8+2 & -2+16-2&8-4-8 \\8+2+1 & -4-4-1&16+1-4 \\ -8-2+1 &4+4-1 &-16-1-4 \end{bmatrix}\)
\(=\begin{bmatrix}-2 & 12&-4 \\11 & -9&13 \\ -9 &7 &-21 \end{bmatrix} \dots (2)\)
Step 5
From, (1) and (2), it is clear that for any two square matrices A and B of the same size AB=BA is not always true.