question 29 points 1

if the matrix $a = \begin{bmatrix} 1 & 0 & 5 \\ 4 & 5 & 3 \\ 1 & 1 & 2 \end{bmatrix}$, what is the value of $|a|$?

$|a| = 2$
$|a| = 5$

Question

question 29 points 1

if the matrix $a = \begin{bmatrix} 1 & 0 & 5 \\ 4 & 5 & 3 \\ 1 & 1 & 2 \end{bmatrix}$, what is the value of $|a|$?

$|a| = 2$
$|a| = 5$

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": [ "Matrix Determinant", "Determinant Expansion" ], "new_concepts": [], "current_concepts": [ "Matrix Determinant", "Determinant Expansion" ] } ## Identify the matrix and target We are given the $3 \times 3$ matrix: $$ A = \begin{bmatrix} 1 & 0 & 5 \ 4 & 5 & 3 \ 1 & 1 & 2 \end{bmatrix} $$ We need to find the determinant $|A|$. ## Expand along the first row Using the **Matrix Determinant** and **Determinant Expansion** knowledge points $$ \begin{aligned} |A| &= 1 \cdot \begin{vmatrix} 5 & 3 \ 1 & 2 \end{vmatrix} - 0 \cdot \begin{vmatrix} 4 & 3 \ 1 & 2 \end{vmatrix} + 5 \cdot \begin{vmatrix} 4 & 5 \ 1 & 1 \end{vmatrix} \end{aligned} $$ ## Calculate the 2x2 determinants Using the **Matrix Determinant** knowledge point $$ \begin{aligned} \begin{vmatrix} 5 & 3 \ 1 & 2 \end{vmatrix} &= (5)(2) - (3)(1) = 10 - 3 = 7\ \begin{vmatrix} 4 & 5 \ 1 & 1 \end{vmatrix} &= (4)(1) - (5)(1) = 4 - 5 = -1 \end{aligned} $$ ## Compute the final determinant value Using the **Matrix Determinant** knowledge point $$ \begin{aligned} |A| &= 1(7) - 0 + 5(-1)\ |A| &= 7 - 5 = 2 \end{aligned} $$

(A) $|A| = 2$

(B) $|A| = 5$

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

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QUESTION IMAGE

Question

Explanation:

Identify the matrix and target

Expand along the first row

Calculate the 2x2 determinants

Compute the final determinant value

Answer: