consider the system of equations. find the determinant value of the matrix of the coefficients of the variables.

$x + y + z = 4$
$4x + 3y + 3z = 6$
$3y + 3z - 4x = 9$

$|a| = 6$
$|a| = 7$

Question

consider the system of equations. find the determinant value of the matrix of the coefficients of the variables.

$x + y + z = 4$
$4x + 3y + 3z = 6$
$3y + 3z - 4x = 9$

$|a| = 6$
$|a| = 7$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Coefficient Matrix", "Matrix Determinant" ], "new_concepts": [], "current_concepts": [ "Coefficient Matrix", "Matrix Determinant", "System of Linear Equations" ] } ## Set up the coefficient matrix Using the *Coefficient Matrix* knowledge point $$ \begin{aligned} &\text{Given system of equations:}\ &1x + 1y + 1z = 4\ &4x + 3y + 3z = 6\ &-4x + 3y + 3z = 9\ &A = \begin{pmatrix} 1 & 1 & 1 \ 4 & 3 & 3 \ -4 & 3 & 3 \end{pmatrix} \end{aligned} $$ ## Calculate the determinant Using the *Matrix Determinant* knowledge point $$ \begin{aligned} |A| &= 1 \begin{vmatrix} 3 & 3 \ 3 & 3 \end{vmatrix} - 1 \begin{vmatrix} 4 & 3 \ -4 & 3 \end{vmatrix} + 1 \begin{vmatrix} 4 & 3 \ -4 & 3 \end{vmatrix}\ &= 1(9 - 9) - 1(12 - (-12)) + 1(12 - (-12))\ &= 0 - 24 + 24\ &= 0 \end{aligned} $$ ## Analyze the incorrect option Using the *Matrix Determinant* knowledge point $$ \begin{aligned} &\text{If the third equation is misread as } 3x + 3y - 4z = 9:\ &A_{\text{misread}} = \begin{pmatrix} 1 & 1 & 1 \ 4 & 3 & 3 \ 3 & 3 & -4 \end{pmatrix}\ &|A_{\text{misread}}| = 1(-12 - 9) - 1(-16 - 9) + 1(12 - 9) = -21 + 25 + 3 = 7 \end{aligned} $$

|A| = 6

|A| = 7

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Matrix Determinant" ] }

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QUESTION IMAGE

Question

Explanation:

Set up the coefficient matrix

Calculate the determinant

Analyze the incorrect option

Answer: