## Linear Algebra Midterm 1 at the Ohio State University (2/3)

## Problem 571

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 2 and contains Problem 4, 5, and 6.

Check out Part 1 and Part 3 for the rest of the exam problems.

**Problem 4**. Let

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

1 \\

2 \\

3

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

2 \\

-1 \\

4

\end{bmatrix}, \mathbf{b}=\begin{bmatrix}

0 \\

a \\

2

\end{bmatrix}.\]

Find all the values for $a$ so that the vector $\mathbf{b}$ is a linear combination of vectors $\mathbf{a}_1$ and $\mathbf{a}_2$.

**Problem 5**.

Find the inverse matrix of

\[A=\begin{bmatrix}

0 & 0 & 2 & 0 \\

0 &1 & 0 & 0 \\

1 & 0 & 0 & 0 \\

1 & 0 & 0 & 1

\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.

**Problem 6**.

Consider the system of linear equations

\begin{align*}

3x_1+2x_2&=1\\

5x_1+3x_2&=2.

\end{align*}

**(a)** Find the coefficient matrix $A$ of the system.

**(b)** Find the inverse matrix of the coefficient matrix $A$.

**(c)** Using the inverse matrix of $A$, find the solution of the system.

(*Linear Algebra Midterm Exam 1, the Ohio State University*)

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