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Chapter 2: Problem 55

Is the sum of two invertible matrices invertible? Explain why or why not. Illustrate your conclusion with appropriate examples.

Short Answer

Expert verified
No, the sum of two invertible matrices is not necessarily invertible. This conclusion is demonstrated by the example of the identity matrix and its negative, both of which are invertible, but their sum results in the zero matrix, which is non-invertible. In other words, there does not exist another matrix which, when multiplied by the zero matrix, will yield the identity matrix.

Step by step solution

01

Define terms

A square matrix \( A \) is invertible if and only if there exists another square matrix \( B \) such that the multiplication of \( A \) and \( B \) (in any order) yields the identity matrix. In other words, if \( AB = BA = I \), where \( I \) is the identity matrix, \( A \) is defined as an invertible matrix, and \( B \) is its inverse.
02

General conception

The sum of two invertible matrices may or may not be invertible. This cannot be guaranteed for all cases.
03

Provide a counter-example

Consider the 2x2 identity matrix \( I = [[1, 0], [0, 1]] \), which is invertible, and its negative \( -I = [[-1, 0], [0, -1]] \), which is also invertible. However, their sum \( I + (-I) = [[0, 0], [0, 0]] \) is the zero matrix, which is not invertible because it does not have an inverse. This example verifies that the sum of two invertible matrices is not necessarily invertible.

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