Math

QuestionIdentify if the matrices are in reduced echelon form or only echelon form: a. [11011050550003300004]\begin{bmatrix}1 & 1 & 0 & 1 & 1 \\ 0 & 5 & 0 & 5 & 5 \\ 0 & 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 0 & 4\end{bmatrix} b. [101101110000]\begin{bmatrix}1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix} c. [1500001000000001]\begin{bmatrix}1 & 5 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}

Studdy Solution

STEP 1

Assumptions1. We are given three matrices a, b, and c. . We need to determine if each matrix is in reduced echelon form, echelon form only, or neither.

STEP 2

First, let's define what it means for a matrix to be in echelon form and reduced echelon form.
A matrix is in echelon form if1. All nonzero rows are above any rows of all zeros.
2. The leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.

A matrix is in reduced echelon form if1. It is in echelon form.
2. The leading coefficient in each nonzero row is1. . All elements in the column above and below a leading1 are zero.

STEP 3

Let's start with matrix a.
a=[1101105055000330000]a = \left[\begin{array}{lllll}1 &1 &0 &1 &1 \\0 &5 &0 &5 &5 \\0 &0 &0 &3 &3 \\0 &0 &0 &0 &\end{array}\right]We can see that the leading coefficients are1,5,3, and. The leading coefficient in each nonzero row is not1, so it is not in reduced echelon form. However, it is in echelon form because each leading coefficient is strictly to the right of the leading coefficient of the row above it.

STEP 4

Now let's look at matrix b.
b=[101101110000]b = \left[\begin{array}{llll}1 &0 &1 &1 \\0 &1 &1 &1 \\0 &0 &0 &0\end{array}\right]The leading coefficients are1 and1. The leading coefficient in each nonzero row is1, but there are non-zero elements in the column above and below the leading1. So, it is not in reduced echelon form. However, it is in echelon form because each leading coefficient is strictly to the right of the leading coefficient of the row above it.

STEP 5

Finally, let's look at matrix c.
c=[1500001000000001]c = \left[\begin{array}{llll}1 &5 &0 &0 \\0 &0 &1 &0 \\0 &0 &0 &0 \\0 &0 &0 &1\end{array}\right]The leading coefficients are1,1, and1. The leading coefficient in each nonzero row is1, and all elements in the column above and below a leading1 are zero. So, it is in reduced echelon form.
In conclusion, matrix a is in echelon form only, matrix b is in echelon form only, and matrix c is in reduced echelon form.

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