Find the Order of AB and BA
Given:
\[ A = \begin{bmatrix} 2 & 1 & 4 \\ 4 & 1 & 5 \end{bmatrix} \]
\[ B = \begin{bmatrix} 3 & -1 \\ 2 & 2 \\ 1 & 3 \end{bmatrix} \]
Step 1: Determine Order of A and B
\[ A \text{ is of order } 2 \times 3 \]
\[ B \text{ is of order } 3 \times 2 \]
Step 2: Find Order of AB
\[ (2 \times 3)(3 \times 2) \Rightarrow AB \text{ exists and is of order } 2 \times 2 \]
Step 3: Find Order of BA
\[ (3 \times 2)(2 \times 3) \Rightarrow BA \text{ exists and is of order } 3 \times 3 \]
Final Answer:
\[ AB \text{ is of order } 2 \times 2, \quad BA \text{ is of order } 3 \times 3 \]
Conclusion:
Matrix multiplication depends on inner dimensions, and AB and BA generally have different orders.