Evaluate `|(1, 2, 3),(7, 8, 9),(3, 2, -1)|`

Options

A. 12
B. 18
C. 21
D. 24

Answer

To evaluate the determinant of the given 3x3 matrix, we can follow these steps:

Given matrix:
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 7 & 8 & 9 \\ 3 & 2 & -1 \end{pmatrix} \]

### Step 1: Write the determinant formula for a 3x3 matrix
The determinant of a 3x3 matrix can be calculated using the formula:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix:
- \( a = 1, b = 2, c = 3 \)
- \( d = 7, e = 8, f = 9 \)
- \( g = 3, h = 2, i = -1 \)

### Step 2: Substitute the values into the formula
Now, substituting the values into the determinant formula:
\[ |A| = 1(8 \cdot (-1) - 9 \cdot 2) - 2(7 \cdot (-1) - 9 \cdot 3) + 3(7 \cdot 2 - 8 \cdot 3) \]

### Step 3: Calculate each term
1. Calculate \( ei - fh \):
\[ 8 \cdot (-1) - 9 \cdot 2 = -8 - 18 = -26 \]
So, the first term becomes:
\[ 1 \cdot (-26) = -26 \]

2. Calculate \( di - fg \):
\[ 7 \cdot (-1) - 9 \cdot 3 = -7 - 27 = -34 \]
So, the second term becomes:
\[ -2 \cdot (-34) = 68 \]

3. Calculate \( dh - eg \):
\[ 7 \cdot 2 - 8 \cdot 3 = 14 - 24 = -10 \]
So, the third term becomes:
\[ 3 \cdot (-10) = -30 \]

### Step 4: Combine the results
Now, combine all the terms:
\[ |A| = -26 + 68 - 30 \]

### Step 5: Simplify the expression
Calculating this gives:
\[ |A| = 42 - 30 = 12 \]

### Final Answer
Thus, the value of the determinant is:
\[ \boxed{12} \]