Use the following two properties of determinants: |A'| = |A| |AB| = |A||B| Then, |AA'| = |A||A'| = |A|^2 = 8^2 = 64

Explanation:

To find the value of |AA'|, we first need to understand what |A| represents. |A| denotes the determinant of matrix A.

Determinant of a Matrix:
The determinant of a matrix is a scalar value that is computed using the elements of the matrix. It provides important information about the matrix, such as whether the matrix is invertible or singular.

Properties of Determinants:
1. If A is a square matrix, then |A| = |A'|, where A' is the transpose of matrix A.
2. If A and B are square matrices of the same order, then |AB| = |A| * |B|.

Solution:
Given that |A| = 8, we need to find |AA'|.

Step 1:
Let's first calculate the transpose of matrix A.

Transpose of a Matrix:
The transpose of a matrix is obtained by interchanging its rows with columns.

Step 2:
Once we have the transpose of matrix A, we can find the product of A and A'.

Step 3:
Finally, we calculate the determinant of the product matrix to find |AA'|.

Step-by-Step Calculation:

1. Let's assume matrix A is a 2x2 matrix:
A = [a b]
[c d]

2. Since |A| = 8, we can write the determinant of matrix A as:
|A| = ad - bc = 8

3. Transpose of matrix A:
A' = [a c]
[b d]

4. Product of A and A':
AA' = [a b] * [a c] = [a^2 + bc ab + bd]
[c d] [b d] [ac + cd bc + d^2]

5. Determinant of AA' = (a^2 + bc)(bc + d^2) - (ab + bd)(ac + cd)

- Breaking down the determinant:
= a^2bc + a^2d^2 + b^2c^2 + b^2d^3 - a^2bc - abcd - abcd - bcd^2
= a^2d^2 + b^2c^2 - 2abcd - bcd^2

Therefore, the value of |AA'| is a^2d^2 + b^2c^2 - 2abcd - bcd^2.