Inverse Matrix Question

Question

Suppose X and Y are both invertible matrices, and W is a block matrix:

W = [ X Z ]

[ 0 Y ]

Show that W is invertible, and its inverse is:

[ X^-1 -X^-1ZY^-1 ] = W^-1

[ 0 Y ^-1 ]

Answer 1

Hello Eliza- This is a case of block multiplication. The main difference is: when you multiply a block by its inverse, you write the identity matrix I. (Instead of 1, which is what you get when you multiply a scalar by its reciprocal.)

We need to show that the following product is the identity matrix:

[ X Z ] times [X^(-1) -X^(-1)ZY^(-1) ]

[ 0 Y ] [ 0 Y^(-1) ]

It's kind of messy so I'll write it out entry by entry for readability.

The 1,1 entry of the product is X*X^(-1) + Z*0 = I + 0 = I

The 1,2 entry of the product is X*[-X^(-1)ZY^(-1)] + ZY^(-1)

Matrix multiplication is associative, so this simplifies to

[X*(-X)^(-1)]*ZY^(-1) + ZY^(-1) = -I*ZY^(-1) + ZY^(-1) = 0

The 2,1 entry of the product is 0*X^(-1) + Y*0 = 0

The 2,2 entry of the product is 0*[-X^(-1)ZY^(-1)] + Y*Y^(-1) = 0 + I = I

So the product is

[ I 0 ]

[ 0 I }

We have 1's on the diagonal and zeroes everywhere else, so this is the identity matrix.