CFRM 405: Mathematical Methods for Quantitative Finance Homework 4
1. Let
(a) Use elimination to turn A into an upper triangular matrix. How many pivots does A have?
(b) Let b = (1,6,3). Does Ax = b have a solution? (c) Let b = (1,6,5). Does Ax = b have a solution?
(d) Can you find multiple solutions in either part (b) or part (c)? If so, find 2.
(e) Does A have an inverse? Justify your answer using results from this exercise.
2. Suppose AB = I and CA = I where I is the n × n identity matrix.
(a) What are the dimensions of the matrices A, B and C?
(b) Show that B = C.
[Hint: you can write IB = B]
(c) Is A invertible?
3. Let A be a square matrix with the property that A2 = A. Simplify (I −A)2 and (I −A)7.
4. (a) Write the vector (9,2,−5) as a linear combination of the vectors (1,2,3) and (6,4,2) or explain why it can’t be done.
(b) How many pivots does a system of equations with coefficient matrix
have?
5. Suppose A is a 6 × 20 matrix and B is a 20 × 7 matrix.
(a) What are the dimensions of C = AB?
(b) Suppose A, B, and C have been partitioned into block matrices like so:
B11
B12
B21
B22
B31
B32
A11
A12
A13
A21
A22
A23
C11
C12
C21
C22
A =, B = , C =,
Suppose that A11 is 2 × 10, B22 is 4 × 3, and C11 is ? × 4. What are the dimensions of each block of A, B, and C such that all the resulting block matrix multiplications are valid?
[Hint: Make note of every fact you know, sketch all three matrices, and fill in the unknowns step by step]
(c) Write each block of C in terms of blocks of A and B.
6. Let A be an m × n matrix.
(a) The full A = QR factorization contains more information than necessary to reconstruct A. What are the smallest matrices Q˜ and R˜ such that Q˜R˜ = A?
(b) Let A˜ be an m×n matrix (m > n) whose columns each sum to zero, and let A˜ = Q˜R˜ be the reduced QR factorization of A˜. The squared Mahalanobis distance to the point x˜Ti (the ith row of A˜) is
