Math Problem Statement

Find the inverses of matrices A, B, and C. Matrix C is given in block form.

Solution

The problem asks for finding the inverses of matrices AA, BB, and CC, with a hint to use block form for matrix CC.

Matrix AA:

The inverse of matrix AA is partially completed, with a small error in the top left position, and you have filled in some correct values. The goal is to complete this inverse correctly.

Given: A=[−2−2−3−5−6−10112]A = \begin{bmatrix} -2 & -2 & -3 \\ -5 & -6 & -10 \\ 1 & 1 & 2 \end{bmatrix} The provided incomplete inverse A−1A^{-1} has the following structure: A−1=[?1??−1?1??]A^{-1} = \begin{bmatrix} ? & 1 & ? \\ ? & -1 & ? \\ 1 & ? & ? \end{bmatrix} We need to compute the missing entries.

Matrix BB:

The task also involves finding the inverse of matrix BB: B=[−12−45−9151−23]B = \begin{bmatrix} -1 & 2 & -4 \\ 5 & -9 & 15 \\ 1 & -2 & 3 \end{bmatrix} We will need to compute the inverse of this matrix.

Matrix CC:

Matrix CC is given as a block matrix:

A & 0 & 0 \\ 0 & D & 0 \\ 0 & 0 & E \end{bmatrix}$$ Where $$D = \begin{bmatrix} -1 & 2 & -4 \\ 5 & 9 & 15 \end{bmatrix}$$ and $$E = \begin{bmatrix} 1 & -2 \\ 1 & 3 \end{bmatrix}$$. For matrix $$C^{-1}$$, since it's block diagonal, its inverse will also be block diagonal with the inverses of $$A$$, $$D$$, and $$E$$ on the diagonal. Would you like a detailed explanation for calculating the inverses of $$A$$, $$B$$, and the block matrix $$C$$, or should I proceed directly with the solutions? --- ### Questions for further exploration: 1. What is the determinant of matrix $$A$$ and how does it affect its invertibility? 2. What are the steps involved in using the adjugate matrix to find the inverse of a 3x3 matrix? 3. How does block matrix inversion differ from normal matrix inversion? 4. What is the significance of the zeros in block matrices like $$C$$? 5. Can you think of applications where block matrix inverses are particularly useful? ### Tip: When working with matrices, always check the determinant first. If the determinant is zero, the matrix does not have an inverse.

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Math Problem Analysis

Mathematical Concepts

Matrix Inversion
Linear Algebra
Block Matrices

Formulas

Inverse of a matrix: A^{-1} = (1/det(A)) * adj(A)
Block matrix inversion formula

Theorems

Matrix Inversion Theorem
Determinant Theorem

Suitable Grade Level

Undergraduate Level (Linear Algebra Course)

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