Question

Given that \( |AB| = 21 \) and \( |A^{-1}| = 7 \), find \( |B| \).

Hint:

When working with determinants, remember that \( |AB| = |A| \cdot |B| \) and \( |A^{-1}| = \frac{1}{|A|} \).

Answer 1

We are given: \[ |AB| = 21 \quad \text{and} \quad |A^{-1}| = 7 \] Step 1: Use the property of determinants.
The determinant of a product of matrices has the property: \[ |AB| = |A| \cdot |B| \] Thus, from the given information, we have: \[ |A| \cdot |B| = 21 \]
Step 2: Use the property of the inverse matrix.
We also know that the determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix: \[ |A^{-1}| = \frac{1}{|A|} \] From the given, \( |A^{-1}| = 7 \), so: \[ |A| = \frac{1}{7} \]
Step 3: Substitute into the equation.
Substitute \( |A| = \frac{1}{7} \) into the equation \( |A| \cdot |B| = 21 \): \[ \frac{1}{7} \cdot |B| = 21 \]
Step 4: Solve for \( |B| \).
Now, solve for \( |B| \): \[ |B| = 21 \times 7 = 147 \] Thus, the value of \( |B| \) is: \[ \boxed{147} \]