Which of following is true if ` B = [(2,-1,3),(-4,5,1)] and A = [(2,3),(4,-2),(1,5)]` ?

Options

A. Only BA is defined
B. Only AB is defined
C. Both AB and BA are defined
D. Both AB and BA are not not defined

Answer

To determine which statement is true regarding the matrices \( A \) and \( B \), we first need to analyze the dimensions of both matrices and the conditions for matrix multiplication.

### Step 1: Identify the dimensions of matrices \( A \) and \( B \)

Matrix \( B \) is given as:
\[ B = \begin{pmatrix} 2 & -1 & 3 \\ -4 & 5 & 1 \end{pmatrix} \]
This matrix has 2 rows and 3 columns, so its dimension is \( 2 \times 3 \).

Matrix \( A \) is given as:
\[ A = \begin{pmatrix} 2 & 3 \\ 4 & -2 \\ 1 & 5 \end{pmatrix} \]
This matrix has 3 rows and 2 columns, so its dimension is \( 3 \times 2 \).

### Step 2: Check the conditions for matrix multiplication

For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix.

- For \( A \times B \):
- \( A \) has dimensions \( 3 \times 2 \)
- \( B \) has dimensions \( 2 \times 3 \)

Since the number of columns in \( A \) (which is 2) matches the number of rows in \( B \) (which is also 2), the multiplication \( A \times B \) is defined.

- For \( B \times A \):
- \( B \) has dimensions \( 2 \times 3 \)
- \( A \) has dimensions \( 3 \times 2 \)

Again, the number of columns in \( B \) (which is 3) matches the number of rows in \( A \) (which is also 3), so the multiplication \( B \times A \) is also defined.

### Step 3: Conclusion about the multiplication results

Since both \( A \times B \) and \( B \times A \) are defined, we can conclude that:
- \( A \times B \) will yield a \( 3 \times 3 \) matrix.
- \( B \times A \) will yield a \( 2 \times 2 \) matrix.

Thus, the results of \( A \times B \) and \( B \times A \) will be different due to their differing dimensions.

### Final Answer

The correct statement regarding the matrices \( A \) and \( B \) is that both products \( A \times B \) and \( B \times A \) are defined, but they will yield different results.

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