Problem 87 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((A+B)^{-1}=A^{-1}+B^{-1},\) assuming \(A, B,\) and \(A+B\) are invertible.

Step-by-Step Solution

Answer

The statement is False. There's no corrected expression for the given equation on general grounds, unless additional conditions are specified.

Step 1: Understand the problem

The problem gives two operations on invertible matrices \(A\) and \(B\): The first operation sum of the matrices and then inversion of the sum i.e. \((A+B)^{-1}\); And the second operation, inverting the individual matrices first and then summing them i.e. \(A^{-1} + B^{-1}\). Here, we need to determine whether both operations yield the same result.

Step 2: Evaluate both sides of the equation

If \(A,\) \(B,\) and \(A+B\) are invertible, \((A+B)^{-1}\) and \(A^{-1} + B^{-1}\) can be computed. However, it is widely known from the properties of matrix inversions that the inverse of a sum is not equals to the sum of the inverses. In simpler terms, \((A+B)^{-1}\) does not equal \(A^{-1} + B^{-1}\). Thus, the given statement can be stated to be false.

Step 3: Provide the correct expression by recalling matrix inverse property

One of the commonly used properties of matrix inverse is that the inverse of the product of two matrices equals the product of their inverses in a reverse order, i.e., \((AB)^{-1}=B^{-1}A^{-1}\). But there is no similar property that equates the inverse of a sum of matrices to the sum of their inverses. Hence, there’s no corrected expression for the provided expression, unless additional conditions are specified.