\( \left. \begin{array} { l } { A = \left[ \begin{array} { c c } { - 2 } & { 3 } \\ { 1 } & { 4 } \end{array} \right] \quad B = \left[ \begin{array} { c c c } { 5 } & { 6 } & { - 3 } \\ { 4 } & { 1 } & { 2 } \end{array} \right] } \\ { A B = } \end{array} \right. \)

**Step 1. Verify matrix dimensions** - Matrix \( A \) is a \( 2 \times 2 \) matrix. - Matrix \( B \) is a \( 2 \times 3 \) matrix. Since the number of columns of \( A \) (which is 2) equals the number of rows of \( B \) (which is also 2), the product \( AB \) is defined and will be a \( 2 \times 3 \) matrix. **Step 2. Write the matrices** \[ A = \begin{pmatrix} -2 & 3 \\ 1 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 & -3 \\ 4 & 1 & 2 \end{pmatrix} \] **Step 3. Compute the product \( AB \)** The product \( AB \) is a \( 2 \times 3 \) matrix. The element in the \( i \)-th row and \( j \)-th column of \( AB \) is computed by taking the dot product of the \( i \)-th row of \( A \) with the \( j \)-th column of \( B \). - **Element \((1,1)\):** \[ (-2)(5) + (3)(4) = -10 + 12 = 2 \] - **Element \((1,2)\):** \[ (-2)(6) + (3)(1) = -12 + 3 = -9 \] - **Element \((1,3)\):** \[ (-2)(-3) + (3)(2) = 6 + 6 = 12 \] - **Element \((2,1)\):** \[ (1)(5) + (4)(4) = 5 + 16 = 21 \] - **Element \((2,2)\):** \[ (1)(6) + (4)(1) = 6 + 4 = 10 \] - **Element \((2,3)\):** \[ (1)(-3) + (4)(2) = -3 + 8 = 5 \] **Step 4. Write the final product matrix** \[ AB = \begin{pmatrix} 2 & -9 & 12 \\ 21 & 10 & 5 \end{pmatrix} \]