Expand `|{:(3,-5,4),(7,6,1),(1,2,3):}|`

Options

A. `180`
B. `190`
C. `18`
D. `80`

Answer

To expand the determinant \( | \begin{pmatrix} 3 & -5 & 4 \\ 7 & 6 & 1 \\ 1 & 2 & 3 \end{pmatrix} | \), we can use the method of cofactor expansion. Let's proceed step by step.

### Step 1: Write the determinant
We start with the determinant:

\[ D = \begin{vmatrix} 3 & -5 & 4 \\ 7 & 6 & 1 \\ 1 & 2 & 3 \end{vmatrix} \]

### Step 2: Expand using the first row
We will expand the determinant along the first row:

\[ D = 3 \cdot \begin{vmatrix} 6 & 1 \\ 2 & 3 \end{vmatrix} - (-5) \cdot \begin{vmatrix} 7 & 1 \\ 1 & 3 \end{vmatrix} + 4 \cdot \begin{vmatrix} 7 & 6 \\ 1 & 2 \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants
Now we need to calculate each of the 2x2 determinants:

1. For \( \begin{vmatrix} 6 & 1 \\ 2 & 3 \end{vmatrix} \):
\[ = (6 \cdot 3) - (1 \cdot 2) = 18 - 2 = 16 \]

2. For \( \begin{vmatrix} 7 & 1 \\ 1 & 3 \end{vmatrix} \):
\[ = (7 \cdot 3) - (1 \cdot 1) = 21 - 1 = 20 \]

3. For \( \begin{vmatrix} 7 & 6 \\ 1 & 2 \end{vmatrix} \):
\[ = (7 \cdot 2) - (6 \cdot 1) = 14 - 6 = 8 \]

### Step 4: Substitute back into the determinant
Now substitute these values back into our expression for \( D \):

\[ D = 3 \cdot 16 + 5 \cdot 20 + 4 \cdot 8 \]

### Step 5: Calculate the final value
Now we calculate each term:

1. \( 3 \cdot 16 = 48 \)
2. \( 5 \cdot 20 = 100 \)
3. \( 4 \cdot 8 = 32 \)

Now, add these values together:

\[ D = 48 + 100 + 32 = 180 \]

### Final Result
Thus, the value of the determinant is:

\[ \boxed{180} \]