Question:
If \( A \) is a non-singular matrix, then which of the following is not true ?
If \( A \) is a non-singular matrix, then which of the following is not true ?
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If a matrix \( A \) is invertible (non-singular), all related structural matrices such as \( A^{-1} \), \( \text{adj } A \), and \( A^T \) are also fully invertible (non-singular).
Updated On: Jul 6, 2026
- \( \text{adj } A \) is singular
- \( (\text{adj } A)^{-1} = (\text{adj } A^{-1}) \)
- \( |A| \neq 0 \)
- \( A^{-1} \) exists
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The Correct Option is A
Solution and Explanation
Concept:
A square matrix \( A \) of order \( n \) is classified as non-singular if its determinant value is non-zero, i.e., \( |A| \neq 0 \). For any non-singular matrix, its inverse exists and is explicitly given by \( A^{-1} = \frac{\text{adj } A}{|A|} \). Additionally, a known property for determinants of adjoint matrices states that \( |\text{adj } A| = |A|^{n-1} \).
Step 1: Analyze statements (C) and (D). By direct definition, a non-singular matrix satisfies \( |A| \neq 0 \). Therefore, statement (C) is completely true. Furthermore, because \( |A| \neq 0 \), the inverse matrix \( A^{-1} \) exists. This renders statement (D) entirely true as well.
Step 2: Analyze statement (B). Using classical algebraic properties of matrices, taking the inverse of an adjoint yields the exact same result as finding the adjoint of the inverse matrix, meaning \( (\text{adj } A)^{-1} = \text{adj}(A^{-1}) \). This statement is also true.
Step 3: Analyze statement (A). Let's find the determinant value of the adjoint matrix \( \text{adj } A \): \[ |\text{adj } A| = |A|^{n-1} \] Since \( A \) is given to be non-singular, we know that \( |A| \neq 0 \). Consequently, \( |A|^{n-1} \neq 0 \), which implies that \( |\text{adj } A| \neq 0 \). By definition, a matrix with a non-zero determinant is non-singular. Therefore, \( \text{adj } A \) must be non-singular. Statement (A) claims that \( \text{adj } A \) is singular, making it false.
Step 1: Analyze statements (C) and (D). By direct definition, a non-singular matrix satisfies \( |A| \neq 0 \). Therefore, statement (C) is completely true. Furthermore, because \( |A| \neq 0 \), the inverse matrix \( A^{-1} \) exists. This renders statement (D) entirely true as well.
Step 2: Analyze statement (B). Using classical algebraic properties of matrices, taking the inverse of an adjoint yields the exact same result as finding the adjoint of the inverse matrix, meaning \( (\text{adj } A)^{-1} = \text{adj}(A^{-1}) \). This statement is also true.
Step 3: Analyze statement (A). Let's find the determinant value of the adjoint matrix \( \text{adj } A \): \[ |\text{adj } A| = |A|^{n-1} \] Since \( A \) is given to be non-singular, we know that \( |A| \neq 0 \). Consequently, \( |A|^{n-1} \neq 0 \), which implies that \( |\text{adj } A| \neq 0 \). By definition, a matrix with a non-zero determinant is non-singular. Therefore, \( \text{adj } A \) must be non-singular. Statement (A) claims that \( \text{adj } A \) is singular, making it false.
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