We need to determine which provided statement about invertible matrices is not correct. Let's analyze each of the given options:
- \(adjA = |A|A^{-1}\): This is the formula for the adjugate (or adjoint) of the matrix \( A \). This statement is correct. For an invertible matrix, the adjugate is indeed the product of the determinant and the inverse.
- \((A+B)^{-1} = A^{-1} + B^{-1}\): This statement is incorrect. Inverse properties do not linearly distribute over matrix addition. Instead, the inverse of a sum involves a more complex relationship that generally cannot be simplified to such a form.
- \(|A^{-1}| = |A|^{-1}\): This is correct. The determinant of the inverse of a matrix \( A \) is the reciprocal of the determinant of \( A \) itself.
- \((AB)^{-1} = B^{-1}A^{-1}\): This is a well-known property of matrix inverses. For any two invertible matrices \( A \) and \( B \), the inverse of their product is the reverse product of their inverses.
Therefore, the only incorrect statement is \((A+B)^{-1} = A^{-1} + B^{-1}\).
Matrix addition does not have a straightforward rule for inverses just like matrix multiplication or determinants, hence the formula doesn't simplify like this. The calculation or simplification of matrix inverses must adhere to more complex algebraic procedures involving computation of individual inverses and possibly higher-level algebraic transformations or decompositions, especially when dealing with block matrices or systems of linear equations.
Correct Answer: \((A+B)^{-1} = A^{-1} + B^{-1}\)
Important Questions from Matrix Algebra
- If A = \( \left[\begin{array}{cc}0 & 1 \\ −1 & 0\end{array}\right]\) and (aI 2 + bA)2 = A, then
If A = \(\left[\begin{array}{cc}2 & −3 \\3 & 5\end{array}\right]\), then which of the following statements are correct?
A. A is a square matrix
B. A−1 exists
C. A is a symmetric matrix
D. |A| = 19
E. A is a null matrix
Choose the correct answer from the options given below.
If A is Square Matrix of order 3, then product of A and its transpose is
What is the transformation matrix M that transforms a square in the xy-plane defined by (1, 1) T, (-1, 1) T, (-1, -1) T and (1, -1) T to a parallelogram whose corresponding vertices are (2, 1) T, (0, 1) T, (-2, -1) T and (0, -1) T?
Let \(A = \left[ {\begin{array}{*{20}{c}} 1&1&0\\ 0&1&0\\ 1&1&0\\ 0&0&1 \end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}} 1&0&0&0\\ 0&1&1&0\\ 1&0&1&1\\ \end{array}} \right]\) Find the boolean product A ⊙ B of the two matrices.
