Math Problem Statement

q III from it

Suppose A, B, and A+B are invertible matrices. Try to show that C = A^{-1} + B^{-1} is invertible.

Solution

The question labeled III in the image is as follows:

III. Suppose AA, BB, and A+BA+B are invertible matrices. Try to show that C=A1+B1C = A^{-1} + B^{-1} is invertible.


To tackle this question, let’s go through the following steps:

  1. Given Condition: Matrices AA, BB, and A+BA+B are invertible.
  2. Objective: Prove that the matrix C=A1+B1C = A^{-1} + B^{-1} is also invertible.

Solution Outline

  1. Since AA and BB are invertible, it follows that their inverses A1A^{-1} and B1B^{-1} exist.
  2. We can express A+BA + B in terms of AA and BB: (A+B)A1=E+BA1(A + B)A^{-1} = E + BA^{-1}
  3. Further manipulation may reveal an expression for (

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Invertible Matrices

Formulas

(A+B)^{-1} formula
Matrix inversion properties

Theorems

Invertibility of Matrices
Properties of Inverse Matrices

Suitable Grade Level

Undergraduate Level