Show that the block upper triangular matrix A in Example 5 is invertible if and only if...

Show that the block upper triangular matrix A in Example 5 is invertible if and only if both A₁₁ and A₂₂ are invertible. [Hint: If A₁₁ and A₂₂ are invertible, the formula for A^–1 given in Example 5 actually works as the inverse of A.] This fact about A is an important part of several computer algorithms that estimate eigenvalues of matrices. Eigenvalues are discussed in Chapter 5.

Example 5:

A matrix of the form

is said to be block upper triangular. Assume that A₁₁ is p × p, A₂₂ is q × q, and A is invertible. Find a formula for A^–1.