MT132 (M132): Linear Algebra
Tutor Marked Assignment
The TMA covers only chapters 1 and 2. It consists of 4 questions, each question is worth 10 marks. Please solve each question in the space provided. You should give the details of your solutions and not just the final results.
Q−1: Answer each of the following as True or False justifying your answers:
a) [2 marks] If |A| = 1, then AX = O could have more than one solution.
b) [2 marks] Any square matrix A can be written as a sum of symmetric and skew-symmetric matrices.
c) [2 marks] If A is an n×n nonsingular matrix, then A5AT is also nonsingular matrix.
d) [2 marks] If A is an n×n nonsingular matrix such that A-1 = A, then A10 = In.
e) [2 marks] If X1, X2 and X3 are linearly dependent vectors in R3, then X3 is a linear combination of X1 and X2.
Q−2: Consider the linear system: .
a) [3 marks] Find |A|, A is the coefficient matrix for the linear system.
b) [2 marks] If possible, find the inverse of A.
c) [3 marks] Solve the linear system.
d) [2 marks] Change the third equation in the linear system to x y + z = 0. Is the new linear system consistent? Explain your answer.
Q¬−3: Let .
a) [6 marks] Find the matrix A.
b) [4 marks] Find |2A3ATA-1|.
Q−4: Let be a set of vectors in R3.
a) [4 marks] Determine whether the vectors in S are linearly independent.
b) [6 marks] Write, if possible, the vector as a linear combination of the vectors in S.
Tutor Marked Assignment
The TMA covers only chapters 1 and 2. It consists of 4 questions, each question is worth 10 marks. Please solve each question in the space provided. You should give the details of your solutions and not just the final results.
Q−1: Answer each of the following as True or False justifying your answers:
a) [2 marks] If |A| = 1, then AX = O could have more than one solution.
b) [2 marks] Any square matrix A can be written as a sum of symmetric and skew-symmetric matrices.
c) [2 marks] If A is an n×n nonsingular matrix, then A5AT is also nonsingular matrix.
d) [2 marks] If A is an n×n nonsingular matrix such that A-1 = A, then A10 = In.
e) [2 marks] If X1, X2 and X3 are linearly dependent vectors in R3, then X3 is a linear combination of X1 and X2.
Q−2: Consider the linear system: .
a) [3 marks] Find |A|, A is the coefficient matrix for the linear system.
b) [2 marks] If possible, find the inverse of A.
c) [3 marks] Solve the linear system.
d) [2 marks] Change the third equation in the linear system to x y + z = 0. Is the new linear system consistent? Explain your answer.
Q¬−3: Let .
a) [6 marks] Find the matrix A.
b) [4 marks] Find |2A3ATA-1|.
Q−4: Let be a set of vectors in R3.
a) [4 marks] Determine whether the vectors in S are linearly independent.
b) [6 marks] Write, if possible, the vector as a linear combination of the vectors in S.