MA 2071 A '02 Practice Exam 1

  1. Solve the linear system below, using the method of elimination.

    \begin{eqnarray*}
2x+3y & = & 13 \\
x-2y & = & 3 \\
5x + 2y & = & 27
\end{eqnarray*}



  2. Suppose $A$ is a square matrix. Use the properties of matrix operations to show that $A+A^T$ is symmetric. Make sure you note which properties you are using.

  3. Suppose that $A$ is an invertible, square matrix. Use the properties of the inverse to show that the equation $A\mathbf{x} =
\mathbf{b}$ always has a unique solution. Make sure you note which properties you are using.

  4. Solve the linear system below, using the method of Gauss-Jordan reduction. You must show all of your steps in converting the augmented matrix to reduced row echelon form.

    \begin{eqnarray*}
2x +3y+z & = & 1 \\
x+y+z & = & 3 \\
3x+4y+2z & = & 4
\end{eqnarray*}



  5. Solve the homogeneous linear system below, using the method of Gauss-Jordan reduction. You must show all of your steps in converting the augmented matrix to reduced row echelon form.

    \begin{eqnarray*}
x+y-z & = & 0 \\
2x+y+3z & = & 0 \\
\hspace{3em}y-5z & = & 0
\end{eqnarray*}



  6. Show that the matrix

    \begin{displaymath}A = \left[ \begin{array}{cc}
1 & 1 \\
2 & 3
\end{array} \right] \end{displaymath}

    is nonsingular, and compute its inverse. You may use either the formula given in class for a $2 \times 2$ matrix, or the general method involving RREF.

William W. Farr
2002-09-06