the augmented matrix for a linear system is
\\begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 1 \end{bmatrix}\

a. is the matrix in reduced row echelon form?
no yes

b. if the matrix is in reduced row echelon form, find the solution set and enter it below. fill vectors from left to right as needed. leave unneeded vectors blank.

no solution

or

\\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} \quad \\ \quad \\ \quad \end{bmatrix} + r \begin{bmatrix} \quad \\ \quad \\ \quad \end{bmatrix} + s \begin{bmatrix} \quad \\ \quad \\ \quad \end{bmatrix} + t \begin{bmatrix} \quad \\ \quad \\ \quad \end{bmatrix}\

Question

the augmented matrix for a linear system is
\\begin{bmatrix} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 1 \end{bmatrix}\

a. is the matrix in reduced row echelon form?
no yes

b. if the matrix is in reduced row echelon form, find the solution set and enter it below. fill vectors from left to right as needed. leave unneeded vectors blank.

no solution

or

\\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} \quad \\ \quad \\ \quad \end{bmatrix} + r \begin{bmatrix} \quad \\ \quad \\ \quad \end{bmatrix} + s \begin{bmatrix} \quad \\ \quad \\ \quad \end{bmatrix} + t \begin{bmatrix} \quad \\ \quad \\ \quad \end{bmatrix}\

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Reduced Echelon Form" ], "new_concepts": [ "System Solution Set", "Vector Form Solution" ], "current_concepts": [ "Reduced Echelon Form", "System Solution Set", "Vector Form Solution" ] } ## Identify reduced row echelon form Using the *Reduced Echelon Form* knowledge point $$ \begin{aligned} &\text{The matrix is: } \begin{bmatrix} 1 & 0 & 0 & 2 \ 0 & 1 & 0 & 5 \ 0 & 0 & 1 & 1 \end{bmatrix}\ &\text{1. All nonzero rows are above any rows of all zeros (none here).}\ &\text{2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.}\ &\text{3. All entries in a column below a leading entry are zeros.}\ &\text{4. The leading entry in each nonzero row is 1.}\ &\text{5. Each leading 1 is the only nonzero entry in its column.}\ &\text{Thus, the matrix is in reduced row echelon form (RREF).} \end{aligned} $$ ## Solve the linear system Using the *Reduced Echelon Form* knowledge point $$ \begin{aligned} &\text{The augmented matrix corresponds to:}\ &1x_1 + 0x_2 + 0x_3 = 2 \implies x_1 = 2\ &0x_1 + 1x_2 + 0x_3 = 5 \implies x_2 = 5\ &0x_1 + 0x_2 + 1x_3 = 1 \implies x_3 = 1\ &\text{The unique solution vector is: } \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} 2 \ 5 \ 1 \end{bmatrix} \end{aligned} $$ ## Question a Yes ## Question b $$ \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} 2 \ 5 \ 1 \end{bmatrix} $$ (Leave the remaining vectors with parameters $r$, $s$, and $t$ blank) { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Linear Algebra", "Reduced Echelon Form" ] }

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QUESTION IMAGE

Question

Explanation:

Identify reduced row echelon form

Solve the linear system

Answer:

Question a

Question b