Question

If $A$ and $B$ are invertible matrices of same order, then which of the following is not correct?

• A. $A \cdot (\text{adj } A) = (\text{adj } A) \cdot A = |A|I$
• B. $A \cdot \text{adj } A = \text{adj } A \cdot A = |A|$
• C. $(AB)^{-1} = B^{-1} A^{-1}$
• D. $|A| \neq 0, |B| \neq 0$

Hint:

Always perform a "type check" on mathematical equations. The product of two matrices is a matrix. A determinant is a scalar. A matrix cannot equal a scalar. Spotting dimension/type mismatches is a quick way to identify false statements.

Answer 1

The Correct Option is B

To determine which of the given statements about invertible matrices \( A \) and \( B \) is not correct, let's analyze each option:

1. \(A \cdot (\text{adj } A) = (\text{adj } A) \cdot A = |A|I\)

This is an established property of matrices. The product of a matrix \( A \) and its adjugate \((\text{adj } A)\) equals the determinant of \( A \) times the identity matrix of the same order. Thus, this statement is correct.

1. \(A \cdot \text{adj } A = \text{adj } A \cdot A = |A|\)

This statement seems incorrect. In general, the result of \(A \cdot \text{adj } A\) should be \(|A|I\) where \( I \) is the identity matrix of the same order as \( A \), and not merely the scalar \(|A|\). Therefore, this statement does not hold true.

1. \((AB)^{-1} = B^{-1} A^{-1}\)

This is the rule of matrix inversion for the product of two matrices. The inverse of the product of two matrices \( AB \) is given by reversing the order of multiplication and taking the inverse of each matrix individually, which results in \((AB)^{-1} = B^{-1}A^{-1}\). This statement is correct.

1. \(|A| \neq 0, |B| \neq 0\)

This condition simply states that matrices \( A \) and \( B \) are invertible, as their determinants are non-zero. This statement is correct as it aligns with the definition of invertibility in matrices.

Given the analysis, the statement "\(A \cdot \text{adj } A = \text{adj } A \cdot A = |A|\)" is not correct. Therefore, the correct answer is that this statement is incorrect.