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

## Problem 570

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 1 and contains the first three problems.

Check out Part 2 and Part 3 for the rest of the exam problems.

**Problem 1.** Determine all possibilities for the number of solutions of each of the systems of linear equations described below.

**(a)** A consistent system of $5$ equations in $3$ unknowns and the rank of the system is $1$.

**(b)** A homogeneous system of $5$ equations in $4$ unknowns and it has a solution $x_1=1$, $x_2=2$, $x_3=3$, $x_4=4$.

**Problem 2.** Consider the homogeneous system of linear equations whose coefficient matrix is given by the following matrix $A$. Find the vector form for the general solution of the system.

\[A=\begin{bmatrix}

1 & 0 & -1 & -2 \\

2 &1 & -2 & -7 \\

3 & 0 & -3 & -6 \\

0 & 1 & 0 & -3

\end{bmatrix}.\]

**Problem 3.** Let $A$ be the following invertible matrix.

\[A=\begin{bmatrix}

-1 & 2 & 3 & 4 & 5\\

6 & -7 & 8& 9& 10\\

11 & 12 & -13 & 14 & 15\\

16 & 17 & 18& -19 & 20\\

21 & 22 & 23 & 24 & -25

\end{bmatrix}

\]
Let $I$ be the $5\times 5$ identity matrix and let $B$ be a $5\times 5$ matrix.

Suppose that $ABA^{-1}=I$.

Then determine the matrix $B$.

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

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