Question

If $ A \text{ is a matrix of order 4 such that } A(\text{adj } A) = 10 I, \text{ then } |\text{adj } A| \text{ is equal to} $

• A. 10
• B. 100
• C. 1000
• D. 10000

Hint:

For any matrix \( A \), \( A (\text{adj } A) = |A| I \), and the determinant of the adjugate matrix is given by \( |\text{adj } A| = |A|^{n-1} \), where \( n \) is the order of the matrix.

Answer 1

The Correct Option is C

Given the matrix equation:

\[ A (\text{adj } A) = 10I \]

1. Apply Fundamental Matrix Identity:
We know from matrix theory that:
\[ A (\text{adj } A) = |A| I \]

2. Equate Both Expressions:
Comparing with the given equation:
\[ 10I = |A| I \implies |A| = 10 \]

3. Determine Order of Matrix A:
The problem implies A is 4×4 (as evident from the context). For an n×n matrix:
\[ |\text{adj } A| = |A|^{n-1} \]

4. Calculate Adjugate Determinant:
For n = 4:
\[ |\text{adj } A| = 10^{4-1} = 10^3 = 1000 \]

Final Result:
\[ |\text{adj } A| = 1000 \]

Answer 2

Given the equation \( A(\text{adj } A) = 10 I \), we know that: \[ A (\text{adj } A) = |A| I \] where \( |A| \) is the determinant of matrix \( A \). In this case, we are given that: \[ A (\text{adj } A) = 10 I \] which implies: \[ |A| (\text{adj } A) = 10 I \] Since \( \text{adj } A = |A|^{n-1} A^{-1} \) (where \( n \) is the order of the matrix), for a matrix of order 4, \( \text{adj } A = |A|^{3} A^{-1} \). Therefore, \( |\text{adj } A| = |A|^{3} \), and from the given equation, we find: \[ |\text{adj } A| = 1000 \]