Q1. Let the square matrix A be defined in terms of vector v and scalar a as, vvT A =I+α√Tv Show that ATA. (i) (ii) If a = -2, show that A-₁ = A (i.e. A² = I). Q2. If P is a square, non-singular matrix, show that following block matrix is also non-singular, A=P P+P- Lp (Hint: Show that Ax = 0 if and only if x = 0, where x is a block vector.) Q3. Let A, B and C be non-singular square matrices of identical size. Compute the inverse of the block matrices, (i) (ii) LB (Hint: Assume that the inverse is a block matrix, and then multiply the matrix and its inverse. Then solve for each block of the inverse.) Q4. Let A, B and C be non-singular square matrices of identical size. Find a matrix X to satisfy the following equation, I B₁ 0 [B]=[CA] D-CA-+B]}] Q5. Let A and B be non-singular square matrices of identical size. Prove that for any natural number n, (BAB-¹)"BA"B-¹. Q6. Prove the following variant of the Woodbury formula, (A + CBCT)-1=A-1-A-¹C(B-¹ + CTA-¹C)-¹CTA-¹. Q7. Prove the following Searle's identities, (I + A-¹)-1=A(I+A)-¹, (i) (ii) (I + AB)-¹A = A(I + BA)-¹. Q8. Prove that for any N x 1 vector x, X2≤ X ≤ X2 √N (Hint: Στις [laΣ max |a| .) max la√max la,|², Q 9. Given any N x 1 vector x, the Jain's fairness index is defined as, J(x)=(- x 1 In the above expression, (-) is the inner product of the two vector arguments, which is equal to cos² where is the angle between x and 1. Prove (without converting to scalars) that, VJ(x)=2(√(x)11 - /(x)I)x (Hint: ||1|| = √N, (1x)² = (x+1)(1¹x).) Q 10. Prove that for any real, square matrix A, ||A|| = √tr(AAT).