Question

consider this matrix:
\\\

$$\begin{bmatrix} 1 & 0 & 0 & -3 & -4 & 5 \\\\ 0 & 0 & 1 & 2 & 0 & -1 \\end{bmatrix}$$

\\

which of these criteria are satisfied by this matrix?

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

is this matrix in echelon form?

• this matrix is in echelon form
• this matrix is not in echelon form

Explanation:

Identify leading entries

Using the Echelon Form knowledge point

$$ A =
LATEXBLOCK0
$$

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

Evaluate criteria 1 and 2

Using the Reduced Echelon Form knowledge point

• Column 1 has leading entry \(1\) at row 1; other entry is \(0\).
• Column 3 has leading entry \(1\) at row 2; other entry is \(0\).
• Thus, each leading entry is the only nonzero entry in its column.
• Below the leading entry in column 1 is \(0\). There are no entries below the leading entry in column 3.
• Thus, all entries in a column below a leading entry are zeroes.

Evaluate criteria 3, 4, and 5

Using the Echelon Form knowledge point

• There are no rows of all zeroes, so the condition holds vacuously.
• The leading entry of row 2 (column 3) is to the right of the leading entry of row 1 (column 1).
• The leading entry of row 1 is \(1\), and the leading entry of row 2 is \(1\).

Determine echelon form status

Using the Echelon Form knowledge point

• All three defining properties of echelon form are satisfied.
• Therefore, the matrix is in echelon form.

Answer:

Question 1

• Each leading entry is the only nonzero entry in its column (Correct answer)
• All entries in a column below a leading entry are zeroes (Correct answer)
• Any rows of all zeroes are below any other rows (Correct answer)
• Each leading entry of a row is in a column to the right of the leading entry in the row above it (Correct answer)
• The leading entry of each row is 1 (Correct answer)

Question 2

• This matrix is in echelon form (Correct answer)
• This matrix is not in echelon form