A matrix is given.

Question

A matrix is given.

	1	9	0	−1	0
0	0	1	7	0
0	0	0	0	1
0	0	0	0	0

(a) Determine whether the matrix is in row-echelon form.

YesNo    

(b) Determine whether the matrix is in reduced row-echelon form.

YesNo    

(c) Write the system of equations for which the given matrix is the augmented matrix. (Enter each answer in terms of x, y, z, and w.)

	x +			=	0
z +			=	0
		=	1
		=	0

Answer 1

In the given matrix:

[1 9 0 -1 0

0 0 1 7 0

0 0 0 0 1

0 0 0 0 0]

a. Determine whether the matrix is in row reduced echelon form.

1. In order for a matrix to be in row echelon form the following conditions or rules must be satisfied:
2. If there are any rows that consist of all zeros, then they are at the bottom of the matrix
3. For any two successive rows that do not consist of all zeros, the leading variable of the lower row occurs farther to the right of the leading nonzero entry
4. The leading entries must be 1

Looking at the definition of a matrix in row reduced echelon form above, we can say that the first condition is satisfied as the bottom or fourth row contains all 0's and there are no other rows that contain all 0's.

Next, we can conclude that all the leading nonzero entries of the lower row occurs father to the right of the leading variable. The leading variables are in bold below and we can see that the leading variable in the row below the other always occurs farther to the right of the leading variable. Additionally, another way to tell if the leading variable occurs farther to the right of the leading entry is to look at the numbers below the leading variable. And if they are all 0's this means that the condition is satisfied. If not, then the condition is not satisfied.

[1 9 0 -1 0

0 0 1 7 0

0 0 0 0 1

0 0 0 0 0]

Lastly, we can also conclude that the leading entries must be 1. We can see this above in bold.

Since all the conditions are satisfied, the matrix is in row reduced echelon form.

Answer:

Yes

b. Determine whether the matrix is in row reduced echelon form.

1. In order for the matrix to be in row reduced echelon form, the matrix must satisfy all the rules listed above for a matrix to be in row echelon form and:
2. Every column that contains a leading 1 entry of a row has 0's everywhere else in its column.

We can see that this condition is satisfied as all of the columns that contain leading 1's have 0's everywhere along its columns. In bold below, the leading variable 1 and its column that contains 0's everywhere else is shown.

[1 9 0 -1 0

0 0 1 7 0

0 0 0 0 1

0 0 0 0 0]

Since all the conditions are satisfied, the matrix is in row reduced echelon form (RREF).

Answer:

Yes

c. Write the system of equations for which the given matrix is the augmented matrix. (Enter each answer in terms of x, y, z, and w.)

To write the system of equation for the given augmented matrix, we can first assign each column with x, y, z, and w.

x y z w

[1 9 0 -1 0

0 0 1 7 0

0 0 0 0 1

0 0 0 0 0]

Now, we can use the corresponding leading entries and variables that we just assigned to make a system of equations.

Answer:

x + 9y - w = 0

z + 7w = 0

0 = 1

0 = 0

Additionally, the answer to this system of equations is no solution as 0 ≠ 1.