PROF. A. SUCIU

MTH 1230 — LINEAR ALGEBRA — FALL 1997

SAMPLE QUIZ #1

1. Solve the following system by Gaussian elimination in matrix form, with back substitution. Indicate for each step which row operation you use.

−5u + 3v + 4w = −5
10u − 8v − 9w = 5
15u + v + 2w = 31

2. Consider the following system of linear equations:

2x_{1} − x_{2} + x_{3} + 3x_{4} = 7
4x_{1} − 2x_{2} + 3x_{3} + 6x_{4} + 4x_{5} = 17
−2x_{1} + x_{2} + x_{3} − 3x_{4} + 8x_{5} = −1

Identify which variables are basic and which are free. Write down the general solution of the system.

3. Let:

A = [\begin{matrix}5 & 6 \\ −1 & 3 \\ 4 & −2 \end{matrix}] , B = [\begin{matrix}−2 & 0 \\ 4 & 5 \\ 7 & 1 \end{matrix}] .

Compute:

2 · A − 3 · B.

4. Let

A = [\begin{matrix}2 & 3 \\ −1 & 4 \end{matrix}] ,  B = [\begin{matrix}1 & 5 & 2 \\ 0 & 3 & −4 \end{matrix}] ,  C = [3 −1] .

Decide whether the following products are defined or not. If they are, compute them:

A · B, B · A, A · C, C · A, B · C, C · B.

5. Find the inverse of the matrix

A = [\begin{matrix}5 & 3 \\ 6 & 4 \end{matrix}] .

6. Use the Gauss–Jordan method to find the inverse of the matrix

A = [\begin{matrix}1 & 4 & −3 \\ −2 & −7 & 6 \\ 1 & 7 & −2 \end{matrix}] .