Suppose the block matrix A on the left side of (7) is invertible and A11 is invertible....

Suppose the block matrix A on the left side of (7) is invertible and A₁₁ is invertible. Show that the Schur complement S of A₁₁ is invertible. [Hint: The outside factors on the right side of (7) are always invertible. Verify this.] When A and A₁₁ are both invertible, (7) leads to a formula for A^–1, using S^–1, , and the other entries in A.