MATH 2090  Answers to the Review Problems for Exam 2  Fall 2001

1. Consider the system of linear equations \begin{aligned}x_{1} & − & 2x_{2} & + & 3x_{3} & + & x_{4} & = & 6 \\ 2x_{1} & − & x_{2} & + & 3x_{3} & − & x_{4} & = & 3 \\ − x_{1} & + & x_{2} & − & 2x_{3} & & & = & −3 \end{aligned}

Write this system in vector form, find the rank of the coefficient matrix, and find the solution set of the system.

In vector form, the system is [\begin{matrix}1 & −2 & 3 & 1 \\ 2 & −1 & 3 & −1 \\ −1 & 1 & 2 & 0 \end{matrix}] [\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = [\begin{matrix}6 \\ 3 \\ −3 \end{matrix}].

The rank of the coefficient matrix is two. The solution set of the system is {(−s + t, s + t − 3, s, t) : s, t real}.

2. Consider the matrix A = [ \begin{matrix}1 & −1 & 2 \\ 2 & −3 & 4 \\ 1 & −1 & 3 \end{matrix} ] and the constant vector b⃗ = [ \begin{matrix}3 \\ 1 \\ 1 \end{matrix} ].

(a) Use Gauss-Jordan elimination to find the inverse of A.

(b) Use your answer from part (a) to solve the system of equations Ax⃗ = b⃗.

(c) Is the transpose of A non-singular? Explain.

(a) A^{−1} = [\begin{matrix}5 & −1 & −2 \\ 2 & −1 & 0 \\ −1 & 0 & 1 \end{matrix}]  (b) x⃗ = A^{−1}b⃗ = [\begin{matrix}12 \\ 5 \\ −2 \end{matrix}]  (c) Yes, (A^{T})^{−1} = (A^{−1})^{T}.

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

(a) Compute the determinant of A. Does the matrix A have an inverse? Explain.

(b) If B is a 4 × 4 matrix with det(B) = −7, what is the determinant of the product AB? Explain.

(c) Can a system Ax⃗ = b⃗ involving this matrix A, and any constant vector b⃗, be inconsistent? Explain.

(a) det A = −7. Since det A ≠ 0, A is non-singular.

(b) det AB = det A det B = 49.

(c) Since A is non-singular, any system of equations with coefficient matrix A has a unique solution (so is consistent).

4. Consider the homogeneous system of linear equations \begin{aligned}(1 − λ)x_{1} & + & 2x_{2} & + & 6x_{3} & = & 0 \\ & & (2 − λ)x_{2} & + & 3x_{3} & = & 0 \\ & & x_{2} & + & (4 − λ)x_{3} & = & 0 \end{aligned}

(a) Find all values of λ for which this system has infinitely many solutions.

(b) Find the solution set of this homogeneous system when λ = 1.

(a) The system has infinitely many solutions if λ = 1 or λ = 5.

(b) When λ = 1, the solution set is {(s, −3t, t) : s, t real}.