Compute AB and BA

Solution:

Step 1: Check order

\(A\) is \(1 \times 4\), \(B\) is \(4 \times 1\)

• \(AB\): Possible → \(1×1\)
• \(BA\): Possible → \(4×4\)

Step 2: Compute \(AB\)

\[ AB = \begin{bmatrix}1 & -1 & 2 & 3\end{bmatrix} \begin{bmatrix}0 \\ 1 \\ 3 \\ 2\end{bmatrix} \] \[ = 1(0) + (-1)(1) + 2(3) + 3(2) \] \[ = 0 – 1 + 6 + 6 = 11 \] \[ AB = \begin{bmatrix}11\end{bmatrix} \]

Step 3: Compute \(BA\)

\[ BA = \begin{bmatrix} 0 \\ 1 \\ 3 \\ 2 \end{bmatrix} \begin{bmatrix} 1 & -1 & 2 & 3 \end{bmatrix} \] \[ = \begin{bmatrix} 0(1) & 0(-1) & 0(2) & 0(3) \\ 1(1) & 1(-1) & 1(2) & 1(3) \\ 3(1) & 3(-1) & 3(2) & 3(3) \\ 2(1) & 2(-1) & 2(2) & 2(3) \end{bmatrix} \] \[ = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & -1 & 2 & 3 \\ 3 & -3 & 6 & 9 \\ 2 & -2 & 4 & 6 \end{bmatrix} \]

Final Answer:

\[ AB = \begin{bmatrix}11\end{bmatrix} \] \[ BA = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & -1 & 2 & 3 \\ 3 & -3 & 6 & 9 \\ 2 & -2 & 4 & 6 \end{bmatrix} \]