Question

6. Suppose $A_{11}$ is invertible. Find X and Y such that $\begin{bmatrix} A_{11} & A_{12} \ A_{21} & A_{22} \end{bmatrix} = \begin{bmatrix} I & 0 \ X & I \end{bmatrix} \begin{bmatrix} A_{11} & 0 \ 0 & A_{22} - A_{21}A_{11}^{-1}A_{12} \end{bmatrix} \begin{bmatrix} I & Y \ 0 & I \end{bmatrix}$. Show your work. (The matrix $A_{22} - A_{21}A_{11}^{-1}A_{12}$ is called the \"Schur complement\" of $A_{11}$.) (6 pts)

          6. Suppose $A_{11}$ is invertible. Find X and Y such that
$\begin{bmatrix} A_{11} & A_{12} \ A_{21} & A_{22} \end{bmatrix} = \begin{bmatrix} I & 0 \ X & I \end{bmatrix} \begin{bmatrix} A_{11} & 0 \ 0 & A_{22} - A_{21}A_{11}^{-1}A_{12} \end{bmatrix} \begin{bmatrix} I & Y \ 0 & I \end{bmatrix}$.
Show your work. (The matrix $A_{22} - A_{21}A_{11}^{-1}A_{12}$ is called the \"Schur complement\" of $A_{11}$.)
(6 pts)

6. Suppose A11 is invertible. Find X and Y such that
< b m a t r i x >
=
< b m a t r i x >

< b m a t r i x >

< b m a t r i x >.
Show your work. (The matrix A22 - A21A11^-1A12 is called the S̈chur complementöf A11.)
(6 pts)

Added by Julie V.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Adi S

06/20/2023

Step 1

$$\begin{bmatrix} I & 0 \\ X & I \end{bmatrix} \begin{bmatrix} A_{11} & 0 \\ 0 & A_{22} - A_{21}A_{11}^{-1}A_{12} \end{bmatrix} \begin{bmatrix} I & Y \\ 0 & I \end{bmatrix} = \begin{bmatrix} I & 0 \\ X & I \end{bmatrix} \begin{bmatrix} A_{11} & A_{11}Y \\ 0 & Show more…

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6. Suppose $A_{11}$ is invertible. Find $X$ and $Y$ such that $$\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} = \begin{bmatrix} I & 0 \\ X & I \end{bmatrix} \begin{bmatrix} A_{11} & 0 \\ 0 & A_{22} - A_{21}A_{11}^{-1}A_{12} \end{bmatrix} \begin{bmatrix} I & Y \\ 0 & I \end{bmatrix}.$$ Show your work. (The matrix $A_{22} - A_{21}A_{11}^{-1}A_{12}$ is called the "Schur complement" of $A_{11}$.) (6 pts)