Question

If \( A \) is a non-singular matrix, then which of the following is not true ?

• A. \( \text{adj } A \) is singular
• B. \( (\text{adj } A)^{-1} = (\text{adj } A^{-1}) \)
• C. \( |A| \neq 0 \)
• D. \( A^{-1} \) exists

Hint:

Determinant properties to remember:
1. \( |AB| = |A||B| \)
2. \( |\text{adj } A| = |A|^{n-1} \)
3. \( |k A| = k^n |A| \) where \( n \) is the order of the matrix.

Answer 1

The Correct Option is A

Concept:
• A matrix \( A \) is non-singular if its determinant is non-zero, i.e., \( |A| \neq 0 \).
• For any square matrix \( A \) of order \( n \), the inverse \( A^{-1} \) exists if and only if \( A \) is non-singular.
• The determinant of the adjoint is related to the determinant of the matrix by \( |\text{adj } A| = |A|^{n-1} \).

Step 1: Analyze the definition of a non-singular matrix
By definition, if \( A \) is non-singular, then:
\[ |A| \neq 0 \]
This confirms that statement (C) is true.
Since \( |A| \neq 0 \), the inverse of the matrix exists, calculated as \( A^{-1} = \frac{1}{|A|} \text{adj } A \).
This confirms that statement (D) is true.

Step 2: Evaluate the nature of the adjoint matrix
Let \( A \) be of order \( n \).
We use the property: \( |\text{adj } A| = |A|^{n-1} \).
Since \( A \) is non-singular, \( |A| \neq 0 \).
This implies that \( |A|^{n-1} \neq 0 \) (assuming \( n > 1 \)).
Therefore, \( |\text{adj } A| \neq 0 \), which means \( \text{adj } A \) is also a non-singular matrix.
Thus, statement (A), which claims \( \text{adj } A \) is singular, is false.

Step 3: Verify statement (B)
The property \( (\text{adj } A)^{-1} = \text{adj } (A^{-1}) \) is a standard identity for non-singular matrices.
Since the question asks for what is not true, (A) is the correct choice.