Find the inverse of the following matrix, (a) Compute X’X and X’Y. Verify by separate calculations that the (i, j) = (2, 2) element in X’X is the sum of squares of column 2 in X. Verify that the (2,...

Find the inverse of the following matrix,

(a) Compute X’X and X’Y. Verify by separate calculations that the (i, j) = (2, 2) element in X’X is the sum of squares of column 2 in X. Verify that the (2, 3) element is the sum of products between columns 2 and 3 of X. Identify the elements in X’Y in terms of sums of squares or products of the columns of X and Y.

(b) Is X of full column rank? What is the rank of X’X?

(c) Obtain (X’X)⁻¹. What is the rank of (X’X)⁻¹? Verify by matrix multiplication that (X’X)⁻¹X’X = I.

(d) Compute P = X(X’X)⁻¹X and verify by matrix multiplication that P is idempotent. Compute the trace tr(P). What is r(P)?