Question

consider this matrix:
\\\

$$\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 4 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}$$

\\

which of these criteria are satisfied by this matrix?

• any rows of all zeroes are below any other rows
• the leading entry of each row is 1
• each leading entry is the only nonzero entry in its column
• all entries in a column below a leading entry are zeroes
• each leading entry of a row is in a column to the right of the leading entry in the row above it

is this matrix in echelon form, reduced echelon form, or neither?

• this matrix is in echelon form, but not reduced echelon form
• this matrix is in reduced echelon form
• this matrix is in neither echelon form nor reduced echelon form

Explanation:

Identify leading entries of each row

Using the Echelon Form knowledge point

$$ A =
LATEXBLOCK0
$$

• Row 1 leading entry: \(1\) in column 1.
• Row 2 leading entry: \(1\) in column 2.
• Row 3 leading entry: \(4\) in column 2.
• Row 4 leading entry: \(1\) in column 3.

Evaluate the five criteria

Using the Echelon Form and Reduced Echelon Form knowledge points

• Criterion 1 (Any rows of all zeroes are below any other rows): Satisfied vacuously since there are no rows of all zeroes.
• Criterion 2 (The leading entry of each row is 1): Satisfied because the leading entries are \(1\) (Row 1), \(1\) (Row 2), \(4\) (Row 3), and \(1\) (Row 4).
• Criterion 3 (Each leading entry is the only nonzero entry in its column): Not satisfied. In column 2, both Row 2 and Row 3 have nonzero entries (\(1\) and \(4\)).
• Criterion 4 (All entries in a column below a leading entry are zeroes): Not satisfied. Below the leading entry of Row 2 (which is in column 2), there is a nonzero entry \(4\) in Row 3.
• Criterion 5 (Each leading entry of a row is in a column to the right of the leading entry in the row above it): Not satisfied. The leading entry of Row 3 is in column 2, which is not to the right of the leading entry of Row 2 (also in column 2).

Determine matrix classification

Using the Echelon Form and Reduced Echelon Form knowledge points

• Since Criterion 4 and Criterion 5 are violated, the matrix is not in echelon form.
• Since it is not in echelon form, it cannot be in reduced echelon form.
• Therefore, the matrix is in neither form.

Answer:

Question 1

[x] Any rows of all zeroes are below any other rows
[x] The leading entry of each row is 1
[ ] Each leading entry is the only nonzero entry in its column
[ ] All entries in a column below a leading entry are zeroes
[ ] Each leading entry of a row is in a column to the right of the leading entry in the row above it

Question 2

• This matrix is in echelon form, but not reduced echelon form
• This matrix is in reduced echelon form
• This matrix is in neither echelon form nor reduced echelon form (Correct answer)