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.

$\begin{align*} -5u+3v+4w &= -5 \\ 10u-8v-9w &= 5 \\ 15u+v+2w &= 31\end{align*}$

2.

Consider the following system of linear equations:

$\begin{align*} 2x_1-x_2+x_3+3x_4 &= 7 \\ 4x_1-2x_2+3x_3+6x_4+4x_5 &= 17 \\ -2x_1+x_2+x_3-3x_4+8x_5 &= -1\end{align*}$

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

3.

Let:

$\begin{displaymath} A= \bmatrix 5 & 6\\ -1 & 3\\ 4 & -2\endbmatrix, \qquad B= \bmatrix -2 & 0\\ 4 & 5\\ 7 & 1\endbmatrix.\end{displaymath}$

Compute:

$\begin{displaymath} 2\cdot A - 3\cdot B.\end{displaymath}$

4.

Let

$\begin{displaymath} A= \bmatrix 2 & 3\\ -1 & 4\endbmatrix, \qquad B= \bmatrix 1... ...0 & 3 & -4 \endbmatrix, \qquad C= \bmatrix 3 & -1 \endbmatrix .\end{displaymath}$