#### MATH 54

##### Midterm 1 | Fall '14| Wehrheim
1. (a) Expresss the following matrix equation as a linear system for variables $x_i$. $\left[ \begin{array}{c c} 7 & 3 \\ -6 & -3 \end{array} \right] x = \left[ \begin{array}{c} -5 \\ 3 \end{array} \right]$

(b) State what it means for $A = \left[ \begin{array}{c c} 7 & 3 \\ -6 & -3 \end{array} \right]$ to have inverse $B = \left[ \begin{array}{c c} 1 & 1 \\ -2 & -\frac{7}{3} \end{array} \right]$
(Make no calculations here - just algebraic statements.)

(c) Demonstrate how to use a property of the inverse from (b) to find the solution to (a).

(d) For the matrix $A$ from (b), use the facts $A \left[ \begin{array}{c} -1 \\ 1 \end{array} \right] = \left[ \begin{array}{c} -4 \\ 3 \end{array} \right]$ and $A \left[ \begin{array}{c} 1 \\ -2 \end{array} \right] = \left[ \begin{array}{c} 1 \\ 0 \end{array} \right]$ to give a solution of $Ax = \left[ \begin{array}{c} 0 \\ 3 \end{array} \right]$ by superposition (i.e. using algebraic properties of matrix-vector multiplication rather than explicitly solving or computing a product).

(e) Describe the solutions of the following system in parametric vector form. \begin{align} x_1 + x_2 + 2x_3 &= 4 \\ x_2 + x_3 &= 3 \\ -2x_1 - 2x_2 - 4x_3 &= -8 \end{align}

2. (a) Compute or explain why the following expressions are undefined for $A = \left[ \begin{array}{c c c} 2 & 0 & -1 \\ 0 & 3 & 1 \end{array} \right]$. $3A \qquad AA^T \qquad A^TA - AA^T$

(b) Calculate the inverse of $A = \left[ \begin{array}{c c c} 1 & 4 & 6 \\ 0 & 2 & 3 \\ 0 & 0 & 5 \end{array} \right]$ by row reduction.

(c) Give a formula for $A^{-1}$ in terms of cofactors and use it to calculate/check the $(1, 2)$ entry of the result in (b).
(Hint: This entry is $\ne 0$.)

3. (a) State a criterion and use it to decide whether the vectors $\left[ \begin{array}{c} 1 \\ 3 \\ -7 \end{array} \right], \left[ \begin{array}{c} 0 \\ -3 \\ 7 \end{array} \right], \left[ \begin{array}{c} 0 \\ 0 \\ -2 \end{array} \right]$ span $\mathbb{R}^3$.

(b) Use your work in (a) and no further calculation to also decide and explain whether the vectors are linearly dependent.
(If you didn't solve (a), state a criterion for linear dependence and make the calculation here.)

(c) Use the fact that $\left[ \begin{array}{c c c} 1 & -2 & 4 \\ 2 & 0 & -4 \\ 3 & 0 & -6 \end{array} \right] \left[ \begin{array}{c} 3 \\ 1 \\ 0 \end{array} \right] = \left[ \begin{array}{c} 1 \\ 6 \\ 9 \end{array} \right]$ to write $\mathbf{w} = \left[ \begin{array}{c} 1 \\ 6 \\ 9 \end{array} \right]$ as a linear combination of the vectors $v_1 = \left[ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right], v_2 = \left[ \begin{array}{c} -2 \\ 0 \\ 0 \end{array} \right]$, and $v_3 = \left[ \begin{array}{c} 4 \\ -4 \\ -6 \end{array} \right]$.

(d) Decide and explain whether there are weights other than the ones found in (c) that allow to write $\mathbf{w}$ as a linear combination of $\mathbf{v_1, v_2, v_3}$.

4. Give counterexamples or justify the following statements just using definitions and algebra (no theorems).
(a) Suppose $A$ is an $m \times n$ matrix and there exists a matrix $D$ so that $AD = I$. Then the columns of $A$ span $\mathbb{R}^m$.

(b) Suppose $A$ is an $m \times n$ matrix and there exists a matrix $D$ so that $AD = I$. Then solutions to $Ax = b$ are unique.

Can $\text{span} \{ \mathbf{v_1, v_2, v_3} \}$ contain vectors that are not in $\text{span} \{ \mathbf{v_1, v_2, v_3, v_4} \}$?
(Give an example or reasoning.)

(d) Explain what equality of $\text{span} \{ v_1, v_2, v_3 \}$ and $\text{span} \{ v_1, v_2, v_3, v_4 \}$ would imply about linear (in)dependence of the vectors $v_1, v_2, v_3, v_4$?

5. (a) Write down the elementary $3 \times 3$ matrices that represent the following row operations:
• adding six times the second row to the first row: $E_1 =$
• scaling the first row by $\frac{1}{3}: E_2 =$
• interchanging the first and third row: $E_3 =$

(b) With the matrices $E_1, E_2, E_3$ from (a) and $A = \left[ \begin{array}{c c c} 0 & 0 & 2 \\ 0 & -1 & 0 \\ 15 & 0 & 0 \end{array} \right]$ calculate $E_1E_2E_3A$.

(c) With the matrices $E_1, E_2, E_3$ from (a), give and explain simple formulas that relate the following for any $3 \times 3$ matrix $B = [ b_1 b_2 b_3 ]$.
• $V_{123} =$ the volume of the parallelepiped determined by the column vectors of $E_1E_2E_3B$
• $V_{321} =$ the volume of the parallelepiped determined by the column vectors of $E_3E_2E_1B$
• $V =$ the volume of the parallelepiped determined by the vectors $E_1b_1, E_1b_2, E_1b_3$

(d) Let $A, B, C$ be $4 \times 4$ matrices with $\det A = 2, \det B = -1, \det C = 5$. Compute
$\det (BC^{-1}A) =$
$\det (2B) =$
$\det (C^TA) - \det (A^TC) =$