If `A=[{:(,-2,3),(,4,1):}] and B=[{:(,1,2),(,3,5):}]`, find (i) AB (ii) BA
(iii) Is AB=BA?
(iv) Write the conclusion that you draw from the result obtained above in (iii).

Answer

To solve the problem step by step, we will calculate the products of matrices A and B, and then determine if AB equals BA.

Given:
\[ A = \begin{pmatrix} -2 & 3 \\ 4 & 1 \end{pmatrix} \]
\[ B = \begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix} \]

### Step 1: Calculate AB

To find the product \( AB \), we perform the matrix multiplication:

\[ AB = \begin{pmatrix} -2 & 3 \\ 4 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix} \]

Calculating the elements of the resulting matrix:

1. First row, first column:
\[ (-2 \times 1) + (3 \times 3) = -2 + 9 = 7 \]

2. First row, second column:
\[ (-2 \times 2) + (3 \times 5) = -4 + 15 = 11 \]

3. Second row, first column:
\[ (4 \times 1) + (1 \times 3) = 4 + 3 = 7 \]

4. Second row, second column:
\[ (4 \times 2) + (1 \times 5) = 8 + 5 = 13 \]

Thus, we have:
\[ AB = \begin{pmatrix} 7 & 11 \\ 7 & 13 \end{pmatrix} \]

### Step 2: Calculate BA

Now, we calculate the product \( BA \):

\[ BA = \begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix} \begin{pmatrix} -2 & 3 \\ 4 & 1 \end{pmatrix} \]

Calculating the elements of the resulting matrix:

1. First row, first column:
\[ (1 \times -2) + (2 \times 4) = -2 + 8 = 6 \]

2. First row, second column:
\[ (1 \times 3) + (2 \times 1) = 3 + 2 = 5 \]

3. Second row, first column:
\[ (3 \times -2) + (5 \times 4) = -6 + 20 = 14 \]

4. Second row, second column:
\[ (3 \times 3) + (5 \times 1) = 9 + 5 = 14 \]

Thus, we have:
\[ BA = \begin{pmatrix} 6 & 5 \\ 14 & 14 \end{pmatrix} \]

### Step 3: Compare AB and BA

Now we compare the results of \( AB \) and \( BA \):

\[ AB = \begin{pmatrix} 7 & 11 \\ 7 & 13 \end{pmatrix}, \quad BA = \begin{pmatrix} 6 & 5 \\ 14 & 14 \end{pmatrix} \]

Since \( AB \neq BA \), we conclude that:

### Step 4: Conclusion

The matrices \( A \) and \( B \) do not commute because \( AB \neq BA \).