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\\)

Explanation:

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<reasoning>

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
\[

$$\begin{aligned} |A| &= 1(7) - 0 + 5(-1)\\ |A| &= 7 - 5 = 2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(|A| = 2\)</mcq-correct>
<mcq-option>(B) \(|A| = 5\)</mcq-option>
</answer>

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Answer:

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<reasoning>

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
\[

$$\begin{aligned} |A| &= 1(7) - 0 + 5(-1)\\ |A| &= 7 - 5 = 2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(|A| = 2\)</mcq-correct>
<mcq-option>(B) \(|A| = 5\)</mcq-option>
</answer>

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