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:

A non-singular matrix preserves its non-singularity through common transformations like finding its adjoint or its inverse.
If \( |A| \neq 0 \), then every property involving \( |A| \) in the denominator or as a product factor will remain non-zero.

Answer 1

The Correct Option is A

Concept:
• A non-singular matrix \( A \) is a square matrix whose determinant \( |A| \) is not equal to zero.
• The inverse \( A^{-1} \) exists if and only if \( |A| \neq 0 \).
• Property of Adjoint: \( |\text{adj } A| = |A|^{n-1} \), where \( n \) is the order of the matrix.

Step 1: Verify options (C) and (D) based on the definition of non-singularity
Since \( A \) is non-singular:
• By definition, \( |A| \neq 0 \). So, (C) is true.
• Since \( |A| \neq 0 \), the inverse \( A^{-1} = \frac{\text{adj } A}{|A|} \) is defined. So, (D) is true.

Step 2: Analyze the singularity of the adjoint matrix in option (A)
We use the determinant property: \( |\text{adj } A| = |A|^{n-1} \). Given \( A \) is non-singular, \( |A| \neq 0 \). Consequently, \( |A|^{n-1} \) will also be non-zero (for any order \( n \geq 1 \)). Since \( |\text{adj } A| \neq 0 \), \( \text{adj } A \) is a non-singular matrix. Therefore, the statement "\( \text{adj } A \) is singular" is false.

Step 3: Check option (B) for completeness
Using the property \( \text{adj } A = |A| A^{-1} \): \[ (\text{adj } A)^{-1} = (|A| A^{-1})^{-1} = \frac{1}{|A|} (A^{-1})^{-1} = \frac{1}{|A|} A \] Also, \( \text{adj }(A^{-1}) = |A^{-1}| (A^{-1})^{-1} = \frac{1}{|A|} A \). Since both sides equal \( \frac{1}{|A|} A \), (B) is a true statement.