- 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.
- 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?
- Let A be a square matrix with the property that A2 = A. Simplify (I −A)2 and (I −A)7.
- (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?
- 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.
- 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˜Ti (the ith row of A˜) is
whereis a covariance matrix. Compute d2i without inverting a matrix.
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