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: Identify the incorrect statement regarding invertible matrices. An invertible matrix is a square matrix with a non-zero determinant.

Step 3: Analyze each property.

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

The formula for the inverse is \( A^{-1} = \frac{1}{|A|} \text{adj}(A) \). Multiplying by \( |A| \) yields \( |A|A^{-1} = \text{adj}(A) \). This statement is correct.

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

This statement claims the inverse of a sum is the sum of inverses, which is generally false. Consider \( A = I \) and \( B = I \).

LHS = \( (I + I)^{-1} = (2I)^{-1} = \frac{1}{2}I \).

RHS = \( I^{-1} + I^{-1} = I + I = 2I \).

Since LHS \( eq \) RHS, this statement is NOT correct.

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

Given \( AA^{-1} = I \). Taking determinants: \( |AA^{-1}| = |I| \). Using \( |XY| = |X||Y| \), we get \( |A||A^{-1}| = 1 \). Since \( |A| eq 0 \), \( |A^{-1}| = \frac{1}{|A|} = |A|^{-1} \). This statement is correct.

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

This is the standard reversal law for matrix inverses and is a known property.

This statement is correct.

Step 4: Conclusion.

The incorrect statement is \( (A + B)^{-1} = A^{-1} + B^{-1} \).