1. Let
   • 

     

     Use elimination to turn A into an upper triangular matrix. How many pivots does A have?

   • Let b = (1,6,3). Does Ax = b have a solution? (c) Let b = (1,6,5). Does Ax = b have a solution?
   • Can you find multiple solutions in either part (b) or part (c)? If so, find 2.
   • 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.
   • What are the dimensions of the matrices A, B and C?
   • Show that B = C.

[Hint: you can write IB = B]

• 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.

• What are the dimensions of C = AB?
• 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.

• 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?
• 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˜^T_i (the i^th row of A^˜) is

whereis a covariance matrix. Compute d²_i without inverting a matrix.