Math 311-101 Additional Exercises — Summer I, 2012

These are not to be handed in.

Row reduction and determinants

  1. For each matrix below, answer the following questions. \[ A = \left(\begin{array}{cccc} 1 & -1 & 4 & 2\\ -2 & 1 & 0 & 1\\ -1 & 0 & 4 & 3\\ 1 & 1 & 6 &3 \end{array}\right) \qquad B = \left(\begin{array}{cccc} 1 & 0 & -1 & 0\\ 2 & 1 & 0 & 3\\ 0& 1 & 5 & 2\\ -1 & 2 & -2 &1 \end{array}\right) \]
    1. Find the determinant via the method of row operations.
    2. Using the determinant of the matrix, state whether the matrix is invertible. If it is, find its inverse via row reduction; also, find the determinant of its inverse.
    3. Is there a nonzero vector $\mathbf x$ such that the matrix times the vector is $\mathbf 0$.? Why?

  2. For 4×4 matrices, state the elementary matrix corresponding to the given row operation.
    1. $R_3 = R_3+2R_1$
    2. $R_2\leftrightarrow R_4$
    3. $R_1=5R_1$

  3. For the matrix $P$ from the previous problem, use row reduction methods to find $P^{-1}$, and then find $\det(P^{-1})$ via any method. Compare your answer with $\frac{1}{\det(P)}$.

  4. Let $A$ be a 5×5 matrix, and let its columns be ${\mathbf a}_j$. Suppose that $3{\mathbf a}_1 - 2{\mathbf a}_5=\mathbf 0$. Is $A$ invertible? Why?

  5. Let $P=\left(\begin{array}{ccc}1 &2&1\\0&4&1\\1&1&0\end{array}\right)$ and let $Q =\left(\begin{array}{ccc}1 &1&-1\\1&-1&4\\2&0&-3\end{array}\right)$. Find the determinants of $P$ and $Q$ using the methods outlined in class (see also p. 111 in the text). Use these to find $\det(PQ)$.

  6. Consider the system of equations below. Use Cramer's rule to find $x_2$. \[ \begin{gather} x_1+2x_2-x_3=1\\ 3x_1+x_2=-2\\ 2x_1+x_2-5x_3=0 \end{gather} \]

Vector spaces

  1. §3.2 (pp. 141-144): 1(a,d), 5(a,b,c), 9(d).

  2. §3.3 (pp. 154-156): 4(b,c), 6(a,d), 7(a,c).

  3. §3.4 (pp. 161-162): 4, 8(a,b,c), 10

  4. Let $F = [e^x,e^{-x}]$ and let $V = \text{span}(F)$.

    1. Show that $F$ is a basis for $V$. (Of course, we already know that it spans.)

    2. Recall that $\cosh(x) := (e^x+e^{-x})/2$ and $\sinh(x) := (e^x- e^{-x})/2$. Find the coordinate vectors for $[\cosh(x)]_F$, $[\sinh(x)]_F$, and $[3\cosh(x)+2\sinh(x)]_F$.

  5. §3.6 (pp. 180-183): 3(a,b), 4(d,e,f), 10.

  6. Let A be the matrix given below. Find bases for the column space, row space, and null space of A. What is the rank of A? What is the nullity of A? \[ A = \left(\begin{array}{cccc} 1 & -2 & 3 & 3\\ 2 & -5 & 7 & 3\\ -1 & 3 & -4 &3 \end{array}\right) \]

Updated 6/9/2012 (fjn).