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Problem 1. True or false? Explain your reasoning. If false, find a condition on A which makes it
true.
(1) If the columns of A are linearly independent, then so are the rows.
(2) If A is invertible, then so is AT
.
(3) If Ax = b has a unique solution for some b ∈ R
m, then nullity(A) = 0.
(4) If the only eigenvalue of A is 1, then A is diagonalizable.
(5) If A maps orthogonal vectors to orthogonal vectors, then A is orthogonal.
Problem 2. Short answer. The answers to all questions below are a single digit between 0 and 9.
(1) Compute the determinant of.
(2) Suppose 1 is an eigenvalue of A. What number is then an eigenvalue of A + I?
(3) Suppose A is a 7 × 9 matrix whose rank is 4. What is the dimension of the row space of A?
(4) The matrix
λ 1 0 0
0 λ 0 0
0 0 λ 0
0 0 1 λ
has only one eigenvalue. What is its geometric multiplicity?
(5) Suppose
1 2 1
0 1 0
1 5 k is not invertible. What is k?
Problem 3. Find the Jordan normal form of the following matrices:
Problem 4. Find an orthogonal matrix Q such that QT AQ is diagonal for
Problem 5. Consider the matrix
(1) Find 3 linearly independent solutions X1(t), X2(t), X3(t) of the differential equation X0
(t) =AX(t).
(2) Find a particular solution of the form a1X1(t) + a2X2(t) + a3X3(t) when X(0) =.
Problem 6. Find bases for the column space, row space, and null space of the matrix
.
1Problem 7. Let v ∈ R3 and let A be a 3 × 3 matrix. Suppose v, Av, A2v are all nonzero and
A3 = 0.
(1) Show that B = {v, Av, A1v} is linearly independent. Deduce B is a basis for R3.
(2) Calculate [LA]B where LA is the linear map given by LAx = Ax.
Problem 8. Suppose A, B are n × n matrices such that AB = In. Show that BA = In.
2
Problem 1. True or false? Explain your reasoning. If false, find a condition on A which makes it
true.
(1) If the columns of A are linearly independent, then so are the rows.
(2) If A is invertible, then so is AT
.
(3) If Ax = b has a unique solution for some b ∈ R
m, then nullity(A) = 0.
(4) If the only eigenvalue of A is 1, then A is diagonalizable.
(5) If A maps orthogonal vectors to orthogonal vectors, then A is orthogonal.
Problem 2. Short answer. The answers to all questions below are a single digit between 0 and 9.
(1) Compute the determinant of.
(2) Suppose 1 is an eigenvalue of A. What number is then an eigenvalue of A + I?
(3) Suppose A is a 7 × 9 matrix whose rank is 4. What is the dimension of the row space of A?
(4) The matrix
λ 1 0 0
0 λ 0 0
0 0 λ 0
0 0 1 λ
has only one eigenvalue. What is its geometric multiplicity?
(5) Suppose
1 2 1
0 1 0
1 5 k is not invertible. What is k?
Problem 3. Find the Jordan normal form of the following matrices:
Problem 4. Find an orthogonal matrix Q such that QT AQ is diagonal for
Problem 5. Consider the matrix
(1) Find 3 linearly independent solutions X1(t), X2(t), X3(t) of the differential equation X0
(t) =AX(t).
(2) Find a particular solution of the form a1X1(t) + a2X2(t) + a3X3(t) when X(0) =.
Problem 6. Find bases for the column space, row space, and null space of the matrix
.
1Problem 7. Let v ∈ R3 and let A be a 3 × 3 matrix. Suppose v, Av, A2v are all nonzero and
A3 = 0.
(1) Show that B = {v, Av, A1v} is linearly independent. Deduce B is a basis for R3.
(2) Calculate [LA]B where LA is the linear map given by LAx = Ax.
Problem 8. Suppose A, B are n × n matrices such that AB = In. Show that BA = In.
2