Inverse of Matrix Sum Formula: Solving for Upper Triangular Matrices

Thread starter Physics_wiz
Start date Apr 6, 2008
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Inverse Matrix Sum

In summary, the inverse of a matrix sum is the matrix that, when added to the original matrix, results in the identity matrix. Not all matrices have an inverse of their sum, as they must be square matrices with non-zero determinants. The inverse of a matrix sum can be calculated using a specific formula. This calculation is useful in solving linear equations and performing matrix operations. However, there are limitations and restrictions, such as the matrices must be square and have non-zero determinants, and may not exist if the matrices are not of compatible dimensions.

Apr 6, 2008

Physics_wiz

Does anyone know a formula for the inverse of a sum of two upper triangular matrices?

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Apr 7, 2008

HallsofIvy

Science Advisor

Homework Helper

What makes you think there is such a formula?

Have you looked at the inverse of
[tex]\left(\begin{array}{cc}a & b \\ 0 & c\end{array}\right)+ \left(\begin{array}{cc}x & y \\ 0 & z\end{array}\right)[/tex]?

1. What is the inverse of a matrix sum?

The inverse of a matrix sum is the matrix that, when added to the original matrix, results in the identity matrix (a square matrix with 1s on the main diagonal and 0s elsewhere).

2. Can all matrices have an inverse of their sum?

No, not all matrices have an inverse of their sum. In order for a matrix to have an inverse, it must be a square matrix and its determinant must not be equal to 0.

3. How is the inverse of a matrix sum calculated?

The inverse of a matrix sum can be calculated using the formula (A + B)^-1 = A^-1 + B^-1, where A and B are the original matrices and ^-1 denotes the inverse.

4. What is the purpose of finding the inverse of a matrix sum?

The inverse of a matrix sum is useful in solving systems of linear equations, as it allows for the calculation of the coefficients of the equations. It is also important in matrix operations such as matrix division.

5. Are there any limitations or restrictions when calculating the inverse of a matrix sum?

Yes, there are some limitations and restrictions when calculating the inverse of a matrix sum. The matrices involved must be square matrices and have non-zero determinants. Additionally, the inverse of a matrix sum may not exist if the matrices are not of compatible dimensions.