Math

QuestionIdentify if the following matrices are in reduced echelon form, echelon form, or neither: a. [100005000011]\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & 1 & 1\end{bmatrix} b. [101101110000]\begin{bmatrix}1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix} c. [1300001000000001]\begin{bmatrix}1 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}

Studdy Solution

STEP 1

Assumptions1. A matrix is in echelon form if it satisfies the following conditions a. All nonzero rows are above any rows of all zeros. b. 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 (also called row-reduced echelon form) if it satisfies the following conditions a. It is in echelon form. b. The leading coefficient in each nonzero row is1. c. All elements in the column above and below a leading1 are zero.

STEP 2

We will first examine matrix a. The matrix a is[100005000011]\left[\begin{array}{llll}1 &0 &0 &0 \\0 &5 &0 &0 \\0 &0 &1 &1\end{array}\right]

STEP 3

We can see that all nonzero rows are above any rows of all zeros and the leading coefficient of a nonzero row is always strictly to the right of the leading coefficient of the row above it. Therefore, matrix a is in echelon form.

STEP 4

However, the leading coefficient in the second row is, not1. Therefore, matrix a is not in reduced echelon form.

STEP 5

Now, let's examine matrix b. The matrix b is[101101110000]\left[\begin{array}{llll}1 &0 &1 &1 \\0 &1 &1 &1 \\0 &0 &0 &0\end{array}\right]

STEP 6

We can see that all nonzero rows are above any rows of all zeros and the leading coefficient of a nonzero row is always strictly to the right of the leading coefficient of the row above it. Therefore, matrix b is in echelon form.

STEP 7

However, there are non-zero elements in the column above and below a leading1. Therefore, matrix b is not in reduced echelon form.

STEP 8

Finally, let's examine matrix c. The matrix c is[1300001000000001]\left[\begin{array}{llll}1 &3 &0 &0 \\0 &0 &1 &0 \\0 &0 &0 &0 \\0 &0 &0 &1\end{array}\right]

STEP 9

We can see that all nonzero rows are above any rows of all zeros and the leading coefficient of a nonzero row is always strictly to the right of the leading coefficient of the row above it. Therefore, matrix c is in echelon form.

STEP 10

The leading coefficient in each nonzero row is and all elements in the column above and below a leading are zero. Therefore, matrix c 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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