Chapter 10, Problem 20RE

(a)

To determine

To state: the dimension of the matrix.

Answer to Problem 20RE

The dimension of the matrix is $4\times 4$

Explanation of Solution

Given:

  $A=\left[\begin{array}{l}1 8 6 -4\\ 0 1 -3 5\\ 0 0 2 -7\\ 1 1 1 0\end{array}\right]$

Calculation:

Consider the following matrix:

  $A=\left[\begin{array}{l}1 8 6 -4\\ 0 1 -3 5\\ 0 0 2 -7\\ 1 1 1 0\end{array}\right]$

The dimension of a matrix is given by $m\times n$ , where $m$ is the number of rows in the matrix and $n$ is the number of columns in the matrix.

In matrix $A$ , number of rows, $m=4$

Number of columns, $n=4$

Thus, the dimension of the matrix is $4\times 4$

Conclusion:

The dimension of the matrix is $4\times 4$

(b)

To determine

To explain: whether the matrix is in row-echelon form

Answer to Problem 20RE

The matrix is not in row-echelon form

Explanation of Solution

Given:

  $A=\left[\begin{array}{l}1 8 6 -4\\ 0 1 -3 5\\ 0 0 2 -7\\ 1 1 1 0\end{array}\right]$

Calculation:

A matrix that satisfies the following conditions is said to be in row-echelon form:

(1) In each row, the first non-zero number should be $1$ . This is called leading entry

(2) The leading entry in each row is to the right of that in the row immediately above it

(3) Rows whose elements are all zeros are at the bottom of the matrix

(4) All elements above and below the leading entry are $0s$ .

The first three conditions should be satisfied for the matrix to be in row-echelon form and if the fourth condition is also satisfied then the matrix is in reduced row-echelon form.

Consider the following matrix:

  $A=\left[\begin{array}{l}1 8 6 -4\\ 0 1 -3 5\\ 0 0 2 -7\\ 1 1 1 0\end{array}\right]$

As the first non-zero number in the second row is not $1$ , the first condition is not satisfied.

Hence the matrix is not in row-echelon form.

Conclusion:

The matrix is not in row-echelon form

(c)

To determine

To explain: whether the matrix is in reduced row-echelon form

Answer to Problem 20RE

The matrix is not in reduced row-echelon form

Explanation of Solution

Given:

  $A=\left[\begin{array}{l}1 8 6 -4\\ 0 1 -3 5\\ 0 0 2 -7\\ 1 1 1 0\end{array}\right]$

Calculation:

In reduced row-echelon form, leading $1\text{'}s$ have $0s$ above and below them.

Along with this, all the three conditions should also be satisfied.

Consider the following matrix:

  $A=\left[\begin{array}{l}1 8 6 -4\\ 0 1 -3 5\\ 0 0 2 -7\\ 1 1 1 0\end{array}\right]$

Matrix $A$ does not satisfy the condition of reduced row-echelon form.

Hence the matrix is not in reduced row-echelon form.

Conclusion:

The matrix is not in reduced row-echelon form

(d)

To determine

To write: the system of equations

Answer to Problem 20RE

The system of equations for which the matrix $A$ is the augmented matrix is

  $\begin{array}{l}x+8y+6z=-4\\ y-3z=5\\ x+y+z=0\end{array}$

Explanation of Solution

Given:

  $A=\left[\begin{array}{l}1 8 6 -4\\ 0 1 -3 5\\ 0 0 2 -7\\ 1 1 1 0\end{array}\right]$

Calculation:

Consider the following matrix:

  $A=\left[\begin{array}{l}1 8 6 -4\\ 0 1 -3 5\\ 0 0 2 -7\\ 1 1 1 0\end{array}\right]$

Augmented matrix consists of the coefficients and constants of the system of equations.

Thus, the system of equations for which the matrix $A$ is the augmented matrix is,

  $\begin{array}{l}x+8y+6z=-4\\ y-3z=5\\ x+y+z=0\end{array}$

Conclusion:

The system of equations for which the matrix $A$ is the augmented matrix is

  $\begin{array}{l}x+8y+6z=-4\\ y-3z=5\\ x+y+z=0\end{array}$