Compute AB and BA

Solution:

Step 1: Check order

\(A\) is \(2 \times 2\), \(B\) is \(2 \times 3\)

• \(AB\): Possible (2×2 × 2×3 → 2×3)
• \(BA\): Not possible (2×3 × 2×2 ❌ mismatch)

Step 2: Compute \(AB\)

\[ AB = \begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{bmatrix} \] \[ = \begin{bmatrix} 1(1)+(-2)(2) & 1(2)+(-2)(3) & 1(3)+(-2)(1) \\ 2(1)+3(2) & 2(2)+3(3) & 2(3)+3(1) \end{bmatrix} \] \[ = \begin{bmatrix} 1-4 & 2-6 & 3-2 \\ 2+6 & 4+9 & 6+3 \end{bmatrix} = \begin{bmatrix} -3 & -4 & 1 \\ 8 & 13 & 9 \end{bmatrix} \]

Step 3: Compute \(BA\)

Not defined because number of columns of \(B\) ≠ number of rows of \(A\).

Final Answer:

\[ AB = \begin{bmatrix} -3 & -4 & 1 \\ 8 & 13 & 9 \end{bmatrix}, \quad BA \text{ is not defined} \]