Objective: Identify the incorrect property of invertible matrices from a given set of statements.
An invertible matrix is a square matrix with a non-zero determinant.
Analysis of Statements:
1. \( \text{adj}(A) = |A|A^{-1} \):
The inverse of matrix \( A \) is defined as \( A^{-1} = \frac{1}{|A|} \text{adj}(A) \). Multiplying by \( |A| \) (which is non-zero for an invertible matrix), we obtain \( |A|A^{-1} = \text{adj}(A) \). This statement is accurate.
2. \( (A + B)^{-1} = A^{-1} + B^{-1} \):
This statement claims the inverse of a sum equals the sum of the inverses. This is generally false. Consider \( A = I \) and \( B = I \). Both are invertible.
Left-Hand Side (LHS): \( (A + B)^{-1} = (I + I)^{-1} = (2I)^{-1} = \frac{1}{2}I^{-1} = \frac{1}{2}I \).
Right-Hand Side (RHS): \( A^{-1} + B^{-1} = I^{-1} + I^{-1} = I + I = 2I \).
Since LHS \( eq \) RHS, this statement is inaccurate.
3. \( |A^{-1}| = |A|^{-1} \):
Given \( AA^{-1} = I \). Taking the determinant of both sides: \( |AA^{-1}| = |I| \). Using the determinant property \( |XY| = |X||Y| \), we get \( |A||A^{-1}| = 1 \). Since \( |A| eq 0 \), we can divide to get \( |A^{-1}| = \frac{1}{|A|} = |A|^{-1} \). This statement is accurate.
4. \( (AB)^{-1} = B^{-1}A^{-1} \):
This is the standard property for the inverse of a matrix product, also known as the reversal law. This statement is accurate.
Conclusion:
The incorrect statement is \( (A + B)^{-1} = A^{-1} + B^{-1} \).