Linear Algebra

Quiz 1. Gauss-Jordan Elimination / Homogeneous System. Math 2568 Spring 2017.

Introduction to Linear Algebra at the Ohio State University quiz problems and solutions

Problem 262

(a) Solve the following system by transforming the augmented matrix to reduced echelon form (Gauss-Jordan elimination). Indicate the elementary row operations you performed.
\begin{align*}
x_1+x_2-x_5&=1\\
x_2+2x_3+x_4+3x_5&=1\\
x_1-x_3+x_4+x_5&=0
\end{align*}

(b) Determine all possibilities for the solution set of a homogeneous system of $2$ equations in $2$ unknowns that has a solution $x_1=1, x_2=5$.

Add to solve later

Solution.

(a) Solve the following system by Gauss-Jordan elimination

The augmented matrix of the system is
\[ \left[\begin{array}{rrrrr|r}
1 & 1 & 0 & 0 &-1 & 1 \\
0 & 1 & 2 & 1 & 3 & 1 \\
1 & 0 & -1 & 1 & 1 & 0 \\
\end{array} \right].\] We apply elementary row operations as follows.
\begin{align*}
\left[\begin{array}{rrrrr|r}
1 & 1 & 0 & 0 &-1 & 1 \\
0 & 1 & 2 & 1 & 3 & 1 \\
1 & 0 & -1 & 1 & 1 & 0 \\
\end{array} \right] \xrightarrow{R_3-R_1}
\left[\begin{array}{rrrrr|r}
1 & 1 & 0 & 0 &-1 & 1 \\
0 & 1 & 2 & 1 & 3 & 1 \\
0 & -1 & -1 & 1 & 2 & -1 \\
\end{array} \right] \xrightarrow{\substack{R_1-R_2\\ R_3+R_2}}\\[6pt] \left[\begin{array}{rrrrr|r}
1 & 0 & -2 & -1 &-4 & 0 \\
0 & 1 & 2 & 1 & 3 & 1 \\
0 & 0 & 1 & 2 & 5 & 0 \\
\end{array} \right] \xrightarrow{\substack{R_1+2R_3\\ R_2-2R_3}}
\left[\begin{array}{rrrrr|r}
1 & 0 & 0 & 3 & 6 & 0 \\
0 & 1 & 0 & -3 & -7 & 1 \\
0 & 0 & 1 & 2 & 5 & 0 \\
\end{array} \right].
\end{align*}

The last matrix is in reduced row echelon form.
The unknowns $x_1, x_2, x_3$ correspond to the leading $1$’s. Thus they are dependent variables and $x_4$ and $x_5$ are independent (free) variables.
From the last matrix, we see that the solution set is given by
\begin{align*}
x_1&=-3x_4-6x_5\\
x_2&=3x_4+7x_5+1\\
x_3&=-2x_4-5x_5
\end{align*}
for any numbers $x_4, x_5$.

(b) Determine all possibilities for the solution set of a homogeneous system

A homogeneous system is always consistent because it has the zero solution $x_1=0, x_2=0$.
So, we already know the possibilities for the solution set are either a unique solution or infinitely many solutions.

Note that the homogeneous system we are inspecting has a solution $x_1=1, x_2=5$, which is different from the zero solution $x_1=0, x_2=0$. Thus, the homogeneous system has at least two solutions.
Thus, the only possibility is that the homogeneous system has infinitely many solutions.

Comment.

These are Quiz 1 problems for Math 2568 (Introduction to Linear Algebra) at OSU in Spring 2017.

List of Quiz Problems of Linear Algebra (Math 2568) at OSU in Spring 2017

There were 13 weekly quizzes. Here is the list of links to the quiz problems and solutions.

Add to solve later

More from my site

Quiz: Possibilities For the Solution Set of a Homogeneous System of Linear Equations 4 multiple choice questions about possibilities for the solution set of a homogeneous system of linear equations. The solutions will be given after completing all problems. (The Ohio State University, Linear Algebra Exam)
Quiz 2. The Vector Form For the General Solution / Transpose Matrices. Math 2568 Spring 2017. (a) The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. \[ \left[\begin{array}{rrrrr|r} 1 & 0 & -1 & 0 &-2 & 0 \\ 0 & 1 & 2 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ \end{array} \right].\] […]
Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) \[\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right. \] where $a,b,c, d$ […]
Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation (a) Find the inverse matrix of \[A=\begin{bmatrix} 1 & 0 & 1 \\ 1 &0 &0 \\ 2 & 1 & 1 \end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason. (b) Find a nonsingular $2\times 2$ matrix $A$ such that \[A^3=A^2B-3A^2,\] where […]
Quiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix (a) Let $A=\begin{bmatrix} 1 & 3 & 0 & 0 \\ 1 &3 & 1 & 2 \\ 1 & 3 & 1 & 2 \end{bmatrix}$. Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$. (b) Find the rank and nullity of the matrix $A$ in part (a). Solution. (a) […]
Vector Form for the General Solution of a System of Linear Equations Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination). Find the vector form for the general […]
The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns Determine all possibilities for the number of solutions of each of the system of linear equations described below. (a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution. (b) A homogeneous system of $5$ equations in $4$ unknowns and the […]
Linear Algebra Midterm 1 at the Ohio State University (1/3) 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 […]