Fall 2000  Math 2090  Exam 2

1. Consider the homogeneous system of linear equations, with coefficient matrix A:

\begin{aligned}x_{1} & − & 2x_{2} & + & x_{3} & − & x_{4} & = & 0 \\ −x_{1} & + & 2x_{2} & & & + & 2x_{4} & = & 0 \\ x_{1} & − & 2x_{2} & + & 4x_{3} & + & 2x_{4} & = & 0 \end{aligned} A = [\begin{matrix}1 & −2 & 1 & −1 \\ −1 & 2 & 0 & 2 \\ 1 & −2 & 4 & 2 \end{matrix} ]

(a) Find a row echelon matrix that is row equivalent to A, and find the rank of A.

The matrix A is row equivalent to [ \begin{matrix}1 & −2 & 1 & −1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{matrix} ], so rank A = 2.

(b) Find the solution set of this system of linear equations.

The solution set is {(2s + 2t, s, −t, t) | s, t ∈ ℝ}.

(c) For this matrix A and any 3 × 1 constant vector b⃗, can you be certain that the non-homogeneous

system of linear equations A x⃗ = b⃗ is consistent? Explain. You cannot be certain. The system A x⃗ = b⃗ is consistent if rank A = rank A^{#}, where A^{#} = [ A | b⃗]. This need not be the case for every b⃗. For instance, the system A x⃗ = b⃗ is inconsistent if b⃗ = [ 1 1 8 ]^{T}.

2. Let A = [ \begin{matrix}0 & 1 & 2 \\ 1 & 1 & −2 \\ 1 & 0 & −2 \end{matrix} ] and b⃗ = [ \begin{matrix}2 \\ 0 \\ 4 \end{matrix} ]. The inverse of A is A^{−1} = [ \begin{matrix}1 & −1 & 2 \\ 0 & 1 & −1 \\ \frac{1}{2} & −\frac{1}{2} & \frac{1}{2} \end{matrix} ].

(a) Use Gauss-Jordan elimination to calculate the inverse of the matrix A by hand. Row reduce [A | I] until you reach [I | A^{−1}].

(b) Solve the system of linear equations Ax⃗ = b⃗.

The solution is x⃗ = A^{−1}b⃗ = [ 10 − 4 3 ]^{T}.

(c) Does the homogeneous system A^{T} x⃗ = 0 have infinitely many solutions? Explain.

Since A is nonsingular, A^{T} is also nonsingular. So A^{T} x⃗ = 0 has only the trivial solution.

3. Consider the system of linear equations

\begin{aligned}x_{1} & − & x_{2} & & & = & 1 , \\ x_{1} & & & − & x_{3} & = & 2 , \\ & & x_{2} & − & x_{3} & = & k^{2} , \end{aligned}

where k is a constant.

(a) Find all values of k for which this system has infinitely many solutions. The system has infintely many solutions if k = 1 or k = −1.

(b) Find all values of k for which this system has no solution.

The system has no solution k is any number other than 1 or −1.

(c) Are there any values of k for which this system has a unique solution? Explain.

There are not. For instance, the rank of the coefficient matrix is 2, which is less than the number of variables.

4. Consider the matrix A = [ \begin{matrix}3 & 2 & 0 & 1 \\ 2 & 4 & 2 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 \end{matrix} ].

(a) Calculate the determinant of the matrix A.

The determinant of this matrix is −18.

(b) Can the answer to part (a) be used to determine if the matrix A is nonsingular?

Explain.

It can: det A ≠ 0 if and only if A is nonsingular.