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Question: If A = \(\begin{bmatrix} - 2 & 3 & - 1 \\ - 1 & 2 & - 1 \\ - 6 & 9 & - 4 \end{bmatrix}\) and B =\...

If A = $\begin{bmatrix}

  • 2 & 3 & - 1 \
  • 1 & 2 & - 1 \
  • 6 & 9 & - 4 \end{bmatrix}andB= and B =\begin{bmatrix} 1 & 3 & - 1 \ 2 & 2 & - 1 \ 3 & 0 & - 1 \end{bmatrix}$, then
A

A B = B A

B

AB | BA

C

A B = 12\frac{1}{2} B A

D

None of these.

Answer

A B = B A

Explanation

Solution

AB = [−23−1−12−1−69−4]\left[ \begin{array} { l l l } - 2 & 3 & - 1 \\ - 1 & 2 & - 1 \\ - 6 & 9 & - 4 \end{array} \right] [13−122−130−1]\begin{bmatrix} 1 & 3 & - 1 \\ 2 & 2 & - 1 \\ 3 & 0 & - 1 \end{bmatrix}

=$\begin{bmatrix}

  • 2 + 6 - 3 & - 1 + 4 - 3 & - 6 + 18 - 12 \
  • 6 + 6 + 0 & - 3 + 4 + 0 & - 18 + 18 + 0 \ 2 - 3 + 1 & 1 - 2 + 1 & 6 - 9 + 4 \end{bmatrix}$

=[100010001]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} BA = $\begin{bmatrix}

  • 2 & 3 & - 1 \
  • 1 & 2 & - 1 \
  • 6 & 9 & - 4 \end{bmatrix}$

= $\begin{bmatrix}

  • 2 + 6 - 3 & - 4 - 2 + 6 & - 6 + 0 + 6 \ 3 + 6 - 9 & 6 + 4 - 9 & 9 + 0 - 9 \ 6 - 6 + 0 & - 2 - 2 + 4 & - 3 + 0 + 4 \end{bmatrix}==\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$ ⇒ AB = BA