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2.5 Problem 34 and 45, Introduction to Linear Algebra by Gilbert Strang

Gilbert Strang's book, Introduction to Linear Algebra 4th ed., chapter 2 section 5, problem 34 and 45 are following. I think Problem 45 is an extension of Problem 34.

Problem 45 is to find the S after this. Then, we could naturally extends this problem to the next:

45' Elimination for a 2 by 2 block matrix: Find inverse of 2 by 2 block matrix.

I try to find the solution of this problem here.

First, I would like to demonstrate the Equation (1) is not correct in the block matrix. (This is correct when the elements are scalars.)


There is a question that how 1/(AD-BC) is defined since A,B,C, and D are all matrices. Isn't it 1/|AD-BC|? But this doesn't matter anyway. Because,


Equation (2)'s element 1,2 is DB-BD != 0, element 2,1 is -CA+AC != 0. This comes from the matrix multiplication is not commutative. This means these are non zero
element, however, this should be zero. Therefore, Equation(1) is not correct. Many of the relationships held in (scalar) matrix also held block matrix. But,
some of the scalar equations are demand on commutative law. This doesn't work on block matrix case.

Then, let's compute the inverse by the straightforward method, elimination.
Let's test it. T^{-1}T is:


TT^{1} is the same. Fantastic! Of course there is no magic and just as expected, but, this is fun.
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