1. **Problem Statement:** Verify whether the matrix multiplication $AB$ equals $BA$ for the given matrices: $$A = \begin{bmatrix}1 & 3 \\ 4 & 5\end{bmatrix}, \quad B = \begin{bmatrix}5 & 3 \\ 4 & 7\end{bmatrix}$$ 2. **Recall the formula for matrix multiplication:** If $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$ and $B = \begin{bmatrix}e & f \\ g & h\end{bmatrix}$, then $$AB = \begin{bmatrix}ae + bg & af + bh \\ ce + dg & cf + dh\end{bmatrix}$$ 3. **Calculate $AB$:** $$AB = \begin{bmatrix}1 \times 5 + 3 \times 4 & 1 \times 3 + 3 \times 7 \\ 4 \times 5 + 5 \times 4 & 4 \times 3 + 5 \times 7\end{bmatrix} = \begin{bmatrix}5 + 12 & 3 + 21 \\ 20 + 20 & 12 + 35\end{bmatrix} = \begin{bmatrix}17 & 24 \\ 40 & 47\end{bmatrix}$$ 4. **Calculate $BA$:** $$BA = \begin{bmatrix}5 \times 1 + 3 \times 4 & 5 \times 3 + 3 \times 5 \\ 4 \times 1 + 7 \times 4 & 4 \times 3 + 7 \times 5\end{bmatrix} = \begin{bmatrix}5 + 12 & 15 + 15 \\ 4 + 28 & 12 + 35\end{bmatrix} = \begin{bmatrix}17 & 30 \\ 32 & 47\end{bmatrix}$$ 5. **Compare $AB$ and $BA$:** $$AB = \begin{bmatrix}17 & 24 \\ 40 & 47\end{bmatrix} \neq BA = \begin{bmatrix}17 & 30 \\ 32 & 47\end{bmatrix}$$ Since $AB \neq BA$, the matrices $A$ and $B$ do not commute. **Final answer:** $AB \neq BA$.