If `A=[(1,-2,3),(-4,2,5)]_(2xx3)`
and `B=[(2,3),(4,5),(2,1)]` then

Options

A. AB,BA, exist and are equal
B. AB,BA exist but not equal
C. AB exists and BA does not exist
D. AB does not exist and BA exists.

Answer

To solve the problem, we need to analyze the matrices A and B, check their dimensions, and determine if the products AB and BA exist, as well as whether they are equal.

### Step-by-Step Solution:

1. **Identify the dimensions of matrices A and B:**
- Matrix A is given as \( A = \begin{pmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{pmatrix} \), which is a \( 2 \times 3 \) matrix (2 rows and 3 columns).
- Matrix B is given as \( B = \begin{pmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{pmatrix} \), which is a \( 3 \times 2 \) matrix (3 rows and 2 columns).

2. **Check if the product AB exists:**
- The product \( AB \) exists if the number of columns in A equals the number of rows in B.
- A has 3 columns and B has 3 rows, so \( AB \) exists.
- The resulting matrix \( AB \) will have dimensions \( 2 \times 2 \) (the number of rows from A and the number of columns from B).

3. **Calculate the product AB:**
\[ AB = \begin{pmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{pmatrix} \]
- To find the element in the first row and first column of AB:
\[ (1 \cdot 2) + (-2 \cdot 4) + (3 \cdot 2) = 2 - 8 + 6 = 0 \]
- To find the element in the first row and second column of AB:
\[ (1 \cdot 3) + (-2 \cdot 5) + (3 \cdot 1) = 3 - 10 + 3 = -4 \]
- To find the element in the second row and first column of AB:
\[ (-4 \cdot 2) + (2 \cdot 4) + (5 \cdot 2) = -8 + 8 + 10 = 10 \]
- To find the element in the second row and second column of AB:
\[ (-4 \cdot 3) + (2 \cdot 5) + (5 \cdot 1) = -12 + 10 + 5 = 3 \]
- Therefore, the product \( AB \) is:
\[ AB = \begin{pmatrix} 0 & -4 \\ 10 & 3 \end{pmatrix} \]

4. **Check if the product BA exists:**
- The product \( BA \) exists if the number of columns in B equals the number of rows in A.
- B has 2 columns and A has 2 rows, so \( BA \) exists.
- The resulting matrix \( BA \) will have dimensions \( 3 \times 3 \) (the number of rows from B and the number of columns from A).

5. **Calculate the product BA:**
\[ BA = \begin{pmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{pmatrix} \]
- For the first row and first column of BA:
\[ (2 \cdot 1) + (3 \cdot -4) = 2 - 12 = -10 \]
- For the first row and second column of BA:
\[ (2 \cdot -2) + (3 \cdot 2) = -4 + 6 = 2 \]
- For the first row and third column of BA:
\[ (2 \cdot 3) + (3 \cdot 5) = 6 + 15 = 21 \]
- For the second row and first column of BA:
\[ (4 \cdot 1) + (5 \cdot -4) = 4 - 20 = -16 \]
- For the second row and second column of BA:
\[ (4 \cdot -2) + (5 \cdot 2) = -8 + 10 = 2 \]
- For the second row and third column of BA:
\[ (4 \cdot 3) + (5 \cdot 5) = 12 + 25 = 37 \]
- For the third row and first column of BA:
\[ (2 \cdot 1) + (1 \cdot -4) = 2 - 4 = -2 \]
- For the third row and second column of BA:
\[ (2 \cdot -2) + (1 \cdot 2) = -4 + 2 = -2 \]
- For the third row and third column of BA:
\[ (2 \cdot 3) + (1 \cdot 5) = 6 + 5 = 11 \]
- Therefore, the product \( BA \) is:
\[ BA = \begin{pmatrix} -10 & 2 & 21 \\ -16 & 2 & 37 \\ -2 & -2 & 11 \end{pmatrix} \]

6. **Conclusion:**
- The product \( AB \) exists and is \( \begin{pmatrix} 0 & -4 \\ 10 & 3 \end{pmatrix} \).
- The product \( BA \) exists and is \( \begin{pmatrix} -10 & 2 & 21 \\ -16 & 2 & 37 \\ -2 & -2 & 11 \end{pmatrix} \).
- Since the dimensions of \( AB \) and \( BA \) are different, \( AB \neq BA \).