MATH 2090  Review Problems for Exam 2  Fall 2000

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.

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.

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.

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.