Determine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of Systems

Problem 648

Determine whether the following augmented matrices are in reduced row echelon form, and calculate the solution sets of their associated systems of linear equations.

The third matrix is not in reduced echelon form because the bottom-left entry is $1$, not $0$, so we first use elementary row operations to put it in this form.
\begin{align*}
\left[\begin{array}{rr|r} 1 & 2 & 0 \\ 1 & 1 & -1 \end{array} \right] \xrightarrow{R_2 – R_1} \left[\begin{array}{rr|r} 1 & 2 & 0 \\ 0 & -1 & -1 \end{array} \right]\\[6pt]
\xrightarrow{(-1) R_2 } \left[\begin{array}{rr|r} 1 & 2 & 0 \\ 0 & 1 & 1 \end{array} \right] \xrightarrow{R_1 – 2 R_2} \left[\begin{array}{rr|r} 1 & 0 & -2 \\ 0 & 1 & 1 \end{array} \right].
\end{align*}
With the reduced matrix, we can read off the solution $x_1 = -2$ and $x_2 = 1$.

Write a Vector as a Linear Combination of Three Vectors
Write the vector $\begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}$ as a linear combination of the vectors
\[\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \, \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix} , \, \begin{bmatrix} 2 \\ 0 \\ 4 \end{bmatrix}.\]
Solution.
We want to find […]

Solving a System of Linear Equations Using Gaussian Elimination
Solve the following system of linear equations using Gaussian elimination.
\begin{align*}
x+2y+3z &=4 \\
5x+6y+7z &=8\\
9x+10y+11z &=12
\end{align*}
Elementary row operations
The three elementary row operations on a matrix are defined as […]

Find Values of $a$ so that Augmented Matrix Represents a Consistent System
Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations.
\[A= \left[\begin{array}{rrr|r}
1 & 2 & 3 & 4 \\
2 &-1 & -2 & a^2 \\
-1 & -7 & -11 & a
\end{array} \right],\]
where $a$ is a real number. Determine all the […]

Solve a System of Linear Equations by Gauss-Jordan Elimination
Solve the following system of linear equations using Gauss-Jordan elimination.
\begin{align*}
6x+8y+6z+3w &=-3 \\
6x-8y+6z-3w &=3\\
8y \,\,\,\,\,\,\,\,\,\,\,- 6w &=6
\end{align*}
We use the following notation.
Elementary row operations.
The […]

A Condition that a Linear System has Nontrivial Solutions
For what value(s) of $a$ does the system have nontrivial solutions?
\begin{align*}
&x_1+2x_2+x_3=0\\
&-x_1-x_2+x_3=0\\
& 3x_1+4x_2+ax_3=0.
\end{align*}
Solution.
First note that the system is homogeneous and hence it is consistent. Thus if the system has a nontrivial […]

Express a Vector as a Linear Combination of Other Vectors
Express the vector $\mathbf{b}=\begin{bmatrix}
2 \\
13 \\
6
\end{bmatrix}$ as a linear combination of the vectors
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
5 \\
-1
\end{bmatrix},
\mathbf{v}_2=
\begin{bmatrix}
1 \\
2 \\
1
[…]

Solving a System of Linear Equations By Using an Inverse Matrix
Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}
(a) Find the coefficient matrix and its inverse matrix.
(b) Using the inverse matrix, solve the system of linear equations.
(The Ohio […]