Chapter 10.3, Problem 18E

(a).

To determine

Given matrix is in row-echelon form or not.

Answer to Problem 18E

The given matrix is not in row-echelon form.

Explanation of Solution

Given:

Given matrix:

  $\left[\begin{array}{cccccc}1& 3& 0& 1& 0& 0\\ 0& 1& 0& 4& 0& 0\\ 0& 0& 0& 1& 1& 2\\ 0& 0& 0& 1& 0& 0\end{array}\right]$

Concept Used:

Properties of row-echelon form.

Calculation:

The first non-zero number in each row is $1$ which is known as leading entry. The leading entry in each row is to the right hand side of the leading entry in the row above it.

Also, the given matrix has row which consisting entirely zeros is at the bottom of the matrix.

So the given matrix is in row-echelon form as it not fits into criteria of the row-echelon form properties.

Conclusion:

Thus the given matrix is not in row-echelon form.

(b).

To determine

Given matrix is in reduced row-echelon form or not.

Answer to Problem 18E

The matrix is not in reduced row-echelon form.

Explanation of Solution

Given:

Given matrix:

  $\left[\begin{array}{cccccc}1& 3& 0& 1& 0& 0\\ 0& 1& 0& 4& 0& 0\\ 0& 0& 0& 1& 1& 2\\ 0& 0& 0& 1& 0& 0\end{array}\right]$

Concept Used:

Properties of reduced row-echelon form.

Calculation:

Given matrix is as below:

  $\left[\begin{array}{cccccc}1& 3& 0& 1& 0& 0\\ 0& 1& 0& 4& 0& 0\\ 0& 0& 0& 1& 1& 2\\ 0& 0& 0& 1& 0& 0\end{array}\right]$

The leading $1$ in the second row does not have $0$ above it.

Therefore, the matrix is not in reduced row-echelon form.

Conclusion:

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

(c).

To determine

System of equations for which given matrix is the argument matrix.

Answer to Problem 18E

System of equations for the given matrix is:

  $\begin{array}{l}x+3y+w=0\\ y+4w=0\\ w+u=2\\ w=0\end{array}$

Explanation of Solution

Given:

Given matrix:

  $\left[\begin{array}{cccccc}1& 3& 0& 1& 0& 0\\ 0& 1& 0& 4& 0& 0\\ 0& 0& 0& 1& 1& 2\\ 0& 0& 0& 1& 0& 0\end{array}\right]$

Calculation:

Given matrix is as below:

  $\left[\begin{array}{cccccc}1& 3& 0& 1& 0& 0\\ 0& 1& 0& 4& 0& 0\\ 0& 0& 0& 1& 1& 2\\ 0& 0& 0& 1& 0& 0\end{array}\right]$

So the system of equations based on the:

  $\begin{array}{l}x+3y+w=0\\ y+4w=0\\ w+u=2\\ w=0\end{array}$

Conclusion:

Thus system of equations for the given matrix is as above.