Question

If A and B are invertible matrices then which of the following statement is NOT correct?

• A. adjA = |A|A-1
• B. (A + B)-1= A-1 + B-1
• C. |A-1| = |A|-1
• D. (AB)-1 = B{-1}A-1

Hint:

Remember that matrix algebra often differs from scalar algebra. Properties like \((a+b)^{-1} = a^{-1} + b^{-1}\) or \(ab = ba\) do not generally hold for matrices. Be especially skeptical of properties involving addition and inversion/multiplication.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:

We need to identify the incorrect statement among the given properties of invertible matrices. An invertible matrix is a square matrix that has a non-zero determinant.

Step 3: Detailed Explanation:

Let's analyze each statement.

1. \( \text{adj}(A) = |A|A^{-1} \):

The formula for the inverse of a matrix \( A \) is given by: \[ A^{-1} = \frac{1}{|A|} \text{adj}(A) \] Multiplying both sides by \( |A| \) (which is non-zero since \( A \) is invertible), we get: \[ |A|A^{-1} = \text{adj}(A) \] This statement is correct.

2. \( (A + B)^{-1} = A^{-1} + B^{-1} \):

This property states that the inverse of a sum is the sum of the inverses. This is generally not true for matrices. We can show this with a counterexample.

Let \( A = I \) and \( B = I \) (where \( I \) is the identity matrix). Both are invertible.

LHS = \( (A + B)^{-1} = (I + I)^{-1} = (2I)^{-1} = \frac{1}{2}I^{-1} = \frac{1}{2}I = \begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix} \).

RHS = \( A^{-1} + B^{-1} = I^{-1} + I^{-1} = I + I = 2I = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \).

Since LHS \( \neq \) RHS, the statement is NOT correct.

3. \( |A^{-1}| = |A|^{-1} \):

We know that a matrix and its inverse satisfy the relation \( AA^{-1} = I \).

Taking the determinant of both sides: \[ |AA^{-1}| = |I| \] Using the property \( |XY| = |X||Y| \), we get: \[ |A||A^{-1}| = 1 \] Since \( A \) is invertible, \( |A| \neq 0 \). We can divide by \( |A| \): \[ |A^{-1}| = \frac{1}{|A|} = |A|^{-1} \] This statement is correct.

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

This is the well-known reversal law for the inverse of a product of matrices. It is a standard property.

This statement is correct.

Step 4: Final Answer:

The statement that is not correct is \( (A + B)^{-1} = A^{-1} + B^{-1} \).