In class today, we finished the section on partitioning matrices. We verified that that block matrices obey the same rules for matrix algebra. In particular, block matrix multiplication works as scalar multiplication as long as the dimensions are appropriate for the sub-matrices to be multiplied together.
We looked at a couple of examples how we can evaluate the structure of larger matrices using block matrices. For example, we prove that the given matrix:
![A= \left[ \begin{array}{cc}A_{11} & O \\ A_{21} & A_{22} \end{array}\right]](/latexrender/pictures/1b62a68571c9e17e7fed4ad84aad5609.gif "A= \left[ \begin{array}{cc}A_{11} & O \\ A_{21} & A_{22} \end{array}\right]")
is nonsingular iff 
and 
are nonsingular. We derived that:
![A^{-1} = \left[ \begin{array}{cc}A_{11}^{-1} & O \\[2ex] -A_{22}^{-1}A_{21}A_{11}^{-1} & A_{22}^{-1} \end{array}\right]](/latexrender/pictures/0124ace7f8a4bf56a611ee2e3963ec78.gif "A^{-1} = \left[ \begin{array}{cc}A_{11}^{-1} & O \\[2ex] -A_{22}^{-1}A_{21}A_{11}^{-1} & A_{22}^{-1} \end{array}\right]")
I then used a rather manufactured example to demonstrate how we might make use of this fact. Find the inverse of the following matrix:
![A=\left[\begin{array}{cc|ccc} 1 & 2 & 0 & 0 & 0 \\ 2 & 3 & 0 & 0 & 0\\ \hline 4 & 3 & 1 & 0 & 0 \\ 1 & 2 & -2 & 1 & 0 \\ 9 & 2 & 0 & 0 & 1 \end{array}\right]](/latexrender/pictures/86f24539b1429418643907aff7157368.gif "A=\left[\begin{array}{cc|ccc} 1 & 2 & 0 & 0 & 0 \\ 2 & 3 & 0 & 0 & 0\\ \hline 4 & 3 & 1 & 0 & 0 \\ 1 & 2 & -2 & 1 & 0 \\ 9 & 2 & 0 & 0 & 1 \end{array}\right]")
I made this matrix up so that 
and 
are easily calculable and thus, so is 
.
I just started to touch on outer product expansions and so I’ll finish that up next time. The first exam has been scheduled for Monday, February 19. I handed out a review and will post an answer key to the review in Blackboard.
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