Let A be positive definite, and let A(k) be the result of k steps of Gaussian elimination (without pivoting). Partition both A and A(k) at k rows and columns to form                          A =  B C...

Let A be positive definite, and let A(k) be the result of k steps of Gaussian elimination (without pivoting). Partition both A and A(k) at k rows and columns to form

                         A =  B C CT D  and A(k) = B∗ C∗ 0 D∗  .

Show that D∗ = D − CTB−1 C. Where is this matrix found in statistics? If A is positive definite, is pivoting necessary?