is the sum of two invertible matrices invertible

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24-08-2024

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In linear algebra, one common question that arises is whether the sum of two invertible matrices is also invertible. To answer this question, let's delve into the properties of matrices and their inverses.

Understanding Invertible Matrices

An invertible matrix, also known as a non-singular matrix, is a square matrix that has an inverse. This means that if ( A ) is an invertible matrix, there exists another matrix ( B ) such that:

[ A \cdot B = B \cdot A = I ]

where ( I ) is the identity matrix. A matrix is invertible if and only if its determinant is non-zero.

The Sum of Two Invertible Matrices

Let ( A ) and ( B ) be two invertible matrices. We want to determine whether the matrix ( C = A + B ) is also invertible.

Case Analysis

1. Counterexample: It is crucial to note that the sum of two invertible matrices is not guaranteed to be invertible. A simple counterexample can illustrate this:

   Let: [ A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} ] Both ( A ) and ( B ) are invertible. However, [ C = A + B = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} + \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} ] The resulting matrix ( C ) is the zero matrix, which is not invertible.

2. Conclusion from the Counterexample: This example illustrates that it is possible for the sum of two invertible matrices to yield a non-invertible matrix. Thus, the sum of two invertible matrices is not necessarily invertible.

General Statement

While the individual matrices ( A ) and ( B ) are invertible, their sum ( C = A + B ) can vary depending on the specific matrices involved. Therefore, we cannot conclusively state that the sum of two invertible matrices is always invertible.

Summary

In conclusion, the sum of two invertible matrices is not guaranteed to be invertible. It depends on the specific matrices being summed. The key takeaway is that while invertible matrices retain certain properties regarding their individual inverses, their sum does not necessarily uphold the invertibility property.