To solve the determinant of the matrix
$A = \begin{bmatrix} 3 & 1 & 5 \\ 2 & 4 & 2 \\ -4 & 2 & 9 \end{bmatrix}$ we will use two methods:

1. The OBE Method (Elementary Row Operations)
2. The Cofactor Expansion Method

1. Using the OBE Method (Elementary Row Operations)

We perform row operations to convert the matrix into an upper triangular form, where the determinant is simply the product of the diagonal elements.

Step 1: Make the first column triangular

Using row operations:

• Swap $R_1$ and $R_3$ to get a leading coefficient of 1 or -1 for better calculations.

$\begin{bmatrix} -4 & 2 & 9 \\ 2 & 4 & 2 \\ 3 & 1 & 5 \end{bmatrix}$

Since we swapped rows, the determinant changes sign.

Step 2: Eliminate the first column below the leading element

Perform:

• $R_2 \to R_2 - \frac{2}{-4} R_1 = R_2 + \frac{1}{2} R_1$
• $R_3 \to R_3 - \frac{3}{-4} R_1 = R_3 + \frac{3}{4} R_1$

After row operations:

$\begin{bmatrix} -4 & 2 & 9 \\ 0 & 5 & 6.5 \\ 0 & 2.5 & 11.75 \end{bmatrix}$

Step 3: Make the second column triangular

Perform:

• $R_3 \to R_3 - \frac{2.5}{5} R_2$

After row operations:

$\begin{bmatrix} -4 & 2 & 9 \\ 0 & 5 & 6.5 \\ 0 & 0 & 8.5 \end{bmatrix}$

Step 4: Compute the determinant

Since the matrix is now upper triangular, the determinant is the product of the diagonal elements:

$\det(A) = (-4) \times (5) \times (8.5) = -170$

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2. Using the Cofactor Expansion Method

We use cofactor expansion along the first row:

- 1 \begin{vmatrix} 2 & 2 \\ -4 & 9 \end{vmatrix} + 5 \begin{vmatrix} 2 & 4 \\ -4 & 2 \end{vmatrix}$$ #### **Step 1: Compute 2×2 Determinants** $$\begin{vmatrix} 4 & 2 \\ 2 & 9 \end{vmatrix} = (4)(9) - (2)(2) = 36 - 4 = 32$$ $$\begin{vmatrix} 2 & 2 \\ -4 & 9 \end{vmatrix} = (2)(9) - (2)(-4) = 18 + 8 = 26$$ $$\begin{vmatrix} 2 & 4 \\ -4 & 2 \end{vmatrix} = (2)(2) - (4)(-4) = 4 + 16 = 20$$ #### **Step 2: Compute Determinant** $$\det(A) = 3(32) - 1(26) + 5(20)$$ $$= 96 - 26 + 100$$ $$= 170$$ --- ### **Comparison** - **OBE Method** gave $$\det(A) = -170$$ because we swapped rows, which changes the sign. - **Cofactor Method** gave $$\det(A) = 170$$ without row swaps. If we correct for the row swap in OBE, both methods agree: $$\det(A) = 170$$ --- ### **Would you like more details or explanations?** Here are some related questions for further understanding: 1. How does the determinant change when swapping rows in a matrix? 2. Why is the determinant of an upper triangular matrix the product of its diagonal? 3. How does row reduction affect determinant calculations? 4. When is cofactor expansion more efficient than row operations? 5. What are the geometric interpretations of determinants? #### **Tip:** Always check for row swaps in the OBE method, as they affect the determinant’s sign!