Question

question 40 points 2

\\(\

$$\begin{bmatrix} 1 & 2 & 3 \\\\ 4 & 2 & 6 \\\\ 3 & 4 & 1 \\end{bmatrix}$$

\\). check whether the matrix a has an inverse.

no
yes

Explanation:

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<reasoning>

State the condition for matrix invertibility

Using the Matrix Inverse Existence knowledge point
\[
\text{A square matrix } A \text{ has an inverse if and only if } \det(A)
e 0.
\]

Define the matrix and set up the determinant

Using the Matrix Determinant knowledge point
\[
A =

$$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 2 & 6 \\ 3 & 4 & 1 \end{bmatrix}$$

\]
\[
\det(A) =

$$\begin{vmatrix} 1 & 2 & 3 \\ 4 & 2 & 6 \\ 3 & 4 & 1 \end{vmatrix}$$

\]

Calculate the determinant using cofactor expansion

Using the Determinant Expansion knowledge point
\[

$$\begin{aligned} \det(A) &= 1 \begin{vmatrix} 2 & 6 \\ 4 & 1 \end{vmatrix} - 2 \begin{vmatrix} 4 & 6 \\ 3 & 1 \end{vmatrix} + 3 \begin{vmatrix} 4 & 2 \\ 3 & 4 \end{vmatrix} \\ &= 1(2 \cdot 1 - 6 \cdot 4) - 2(4 \cdot 1 - 6 \cdot 3) + 3(4 \cdot 4 - 2 \cdot 3) \\ &= 1(2 - 24) - 2(4 - 18) + 3(16 - 6) \\ &= -22 - 2(-14) + 3(10) \\ &= -22 + 28 + 30 \\ &= 36 \end{aligned}$$

\]

Determine if the inverse exists

Using the Matrix Inverse Existence knowledge point
\[
\det(A) = 36
e 0 \implies A \text{ has an inverse.}
\]
</reasoning>

<answer>
<mcq-option>No</mcq-option>
<mcq-correct>Yes</mcq-correct>
</answer>

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Answer:

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<reasoning>

State the condition for matrix invertibility

Using the Matrix Inverse Existence knowledge point
\[
\text{A square matrix } A \text{ has an inverse if and only if } \det(A)
e 0.
\]

Define the matrix and set up the determinant

Using the Matrix Determinant knowledge point
\[
A =

$$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 2 & 6 \\ 3 & 4 & 1 \end{bmatrix}$$

\]
\[
\det(A) =

$$\begin{vmatrix} 1 & 2 & 3 \\ 4 & 2 & 6 \\ 3 & 4 & 1 \end{vmatrix}$$

\]

Calculate the determinant using cofactor expansion

Using the Determinant Expansion knowledge point
\[

$$\begin{aligned} \det(A) &= 1 \begin{vmatrix} 2 & 6 \\ 4 & 1 \end{vmatrix} - 2 \begin{vmatrix} 4 & 6 \\ 3 & 1 \end{vmatrix} + 3 \begin{vmatrix} 4 & 2 \\ 3 & 4 \end{vmatrix} \\ &= 1(2 \cdot 1 - 6 \cdot 4) - 2(4 \cdot 1 - 6 \cdot 3) + 3(4 \cdot 4 - 2 \cdot 3) \\ &= 1(2 - 24) - 2(4 - 18) + 3(16 - 6) \\ &= -22 - 2(-14) + 3(10) \\ &= -22 + 28 + 30 \\ &= 36 \end{aligned}$$

\]

Determine if the inverse exists

Using the Matrix Inverse Existence knowledge point
\[
\det(A) = 36
e 0 \implies A \text{ has an inverse.}
\]
</reasoning>

<answer>
<mcq-option>No</mcq-option>
<mcq-correct>Yes</mcq-correct>
</answer>

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