Find the matrix product AB if A = 2 x 5
 4 3 −2 ,B=
 
 
 
 1 x
 −3 2
 −1 2
 
 (b) Find the largest and the smallest possible value of determinant |AB|.

We are given two matrices \( A \) and \( B \) and need to:

1. Find the matrix product \( AB \).
2. Determine the largest and smallest possible values of the determinant \( |AB| \).

### (a) Matrix Multiplication \( AB \)

First, let's define the matrices properly:
- Matrix \( A \) is a \( 2 \times 3 \) matrix:
  \[
  A = \begin{pmatrix} 4 & 3 & -2 \end{pmatrix}
  \]

- Matrix \( B \) is a \( 3 \times 2 \) matrix:
  \[
  B = \begin{pmatrix} 1 & x \\ -3 & 2 \\ -1 & 2 \end{pmatrix}
  \]

To multiply these matrices, the resulting product \( AB \) will be a \( 2 \times 2 \) matrix. The product is calculated as follows:

\[
AB = A \times B = \begin{pmatrix} 4 & 3 & -2 \end{pmatrix} \times \begin{pmatrix} 1 & x \\ -3 & 2 \\ -1 & 2 \end{pmatrix}
\]

Now, let's compute each entry of \( AB \):

\[
AB_{11} = 4(1) + 3(-3) + (-2)(-1) = 4 - 9 + 2 = -3
\]
\[
AB_{12} = 4(x) + 3(2) + (-2)(2) = 4x + 6 - 4 = 4x + 2
\]

So the product matrix \( AB \) is:
\[
AB = \begin{pmatrix} -3 & 4x + 2 \end{pmatrix}
\]

### (b) Determinant \( |AB| \)

Since \( AB \) is a \( 2 \times 2 \) matrix, its determinant can be calculated using the formula:

\[
|AB| = AB_{11} \times AB_{22} - AB_{12} \times AB_{21}
\]

Given that

Find the matrix product AB if A = 2 x 5 4 3 −2 , B = 1 x −3 2 −1 2  (b) Find the largest and the smallest possible value of determinant |AB|.

Learn how to find the matrix product AB and the determinant |AB| for matrices A and B. This problem explores matrix multiplication and determinant calculation with step-by-step instructions. Suitable for Grades 10-12.

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