Question

question 30 points 2

find the determinant of the 3 by 3 matrix shown below:
\\\

$$\begin{vmatrix} 1 & 2 & 5 \\\\ 2 & 1 & 4 \\\\ 3 & 5 & 1 \\end{vmatrix}$$

\\

30

36

Explanation:

Identify the matrix elements

Using the Matrix Representation knowledge point, we write the given \(3 \times 3\) matrix \(A\) as:

$$ A =
LATEXBLOCK0
$$

We need to find its determinant, denoted as:

$$ \det(A) =
LATEXBLOCK1
$$

Apply cofactor expansion

We expand along the first row to calculate the determinant:

$$ \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} $$

This gives:

$$ \det(A) = 1 \cdot
LATEXBLOCK2
- 2 \cdot
LATEXBLOCK3
+ 5 \cdot
LATEXBLOCK4
$$

Calculate \(2 \times 2\) determinants

We evaluate each of the \(2 \times 2\) sub-determinants:

$$
LATEXBLOCK5
= (1)(1) - (4)(5) = 1 - 20 = -19 $$

$$
LATEXBLOCK6
= (2)(1) - (4)(3) = 2 - 12 = -10 $$

$$
LATEXBLOCK7
= (2)(5) - (1)(3) = 10 - 3 = 7 $$

Combine the terms

We substitute these values back into our expansion formula:

$$ \det(A) = 1(-19) - 2(-10) + 5(7) $$

$$ \det(A) = -19 + 20 + 35 $$

$$ \det(A) = 1 + 35 = 36 $$

Answer:

• 30
• 36 (Correct answer)