Math Problem Statement
Solution
This problem involves proving the inverse of a block matrix, given that and are nonsingular (invertible) matrices. The problem states:
Given matrices and as nonsingular (non-zero determinants), we need to show that:
Solution Outline
For a block matrix of the form , where and are invertible, the inverse can be derived using the formula:
To verify this, we can multiply and to check if we get the identity matrix:
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Calculate :
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Multiply the matrices component-wise:
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The top-left block:
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The top-right block:
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The bottom-left block:
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The bottom-right block:
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Result: After performing these calculations, we find that the product is indeed the identity matrix , thus confirming that
This completes the proof.
Would you like additional details on any part of the calculation?
Related Questions
- What conditions are necessary for a block matrix to have an inverse?
- How does the presence of zero entries in the matrix simplify the inversion process?
- What changes in the formula if ?
- Can we generalize this inversion technique for other block matrix forms?
- What is the practical significance of inverting block matrices in applications?
Tip
For large matrices, breaking them into blocks can simplify computations, especially in cases where some blocks are zeros.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Block Matrix
Formulas
Block matrix inverse formula: \( \begin{pmatrix} A & 0 \\ C & B \end{pmatrix}^{-1} = \begin{pmatrix} A^{-1} & 0 \\ -B^{-1}CA^{-1} & B^{-1} \end{pmatrix} \)
Theorems
Inverse of block matrices
Suitable Grade Level
Undergraduate Level
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