## Linear Algebra Midterm 1 at the Ohio State University (3/3)

## Problem 572

The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.

There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).

The time limit was 55 minutes.

This post is Part 3 and contains Problem 7, 8, and 9.

Check out Part 1 and Part 2 for the rest of the exam problems.

**Problem 7**. Let $A=\begin{bmatrix}

-3 & -4\\

8& 9

\end{bmatrix}$ and $\mathbf{v}=\begin{bmatrix}

-1 \\

2

\end{bmatrix}$.

**(a)** Calculate $A\mathbf{v}$ and find the number $\lambda$ such that $A\mathbf{v}=\lambda \mathbf{v}$.

**(b)** Without forming $A^3$, calculate the vector $A^3\mathbf{v}$.

**Problem 8**. Prove that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.

**Problem 9**.

Determine whether each of the following sentences is true or false.

**(a)** There is a $3\times 3$ homogeneous system that has exactly three solutions.

**(b)** If $A$ and $B$ are $n\times n$ symmetric matrices, then the sum $A+B$ is also symmetric.

**(c)** If $n$-dimensional vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are linearly dependent, then the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$ is also linearly dependent for any $n$-dimensional vector $\mathbf{v}_4$.

**(d)** If the coefficient matrix of a system of linear equations is singular, then the system is inconsistent.

**(e)** The vectors

\[\mathbf{v}_1=\begin{bmatrix}

1 \\

0 \\

1

\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}

0 \\

0 \\

1

\end{bmatrix}\]
are linearly independent.