The value of `|U|` where `U= [(2,3,4),(3,4,5),(4,5,6)]`

Options

A. 3
B. -3
C. 0
D. 2

Answer

To find the value of the determinant \( |U| \) where

\[ U = \begin{pmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{pmatrix} \]

we will follow these steps:

### Step 1: Write down the determinant formula
The determinant of a 3x3 matrix

\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

is calculated using the formula:

\[ |U| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

### Step 2: Substitute the values from matrix \( U \)
For our matrix \( U \):

- \( a = 2, b = 3, c = 4 \)
- \( d = 3, e = 4, f = 5 \)
- \( g = 4, h = 5, i = 6 \)

Substituting these values into the determinant formula gives:

\[ |U| = 2(4 \cdot 6 - 5 \cdot 5) - 3(3 \cdot 6 - 5 \cdot 4) + 4(3 \cdot 5 - 4 \cdot 4) \]

### Step 3: Calculate the individual products
Now we calculate each term:

1. \( 4 \cdot 6 = 24 \)
2. \( 5 \cdot 5 = 25 \)
3. \( 3 \cdot 6 = 18 \)
4. \( 5 \cdot 4 = 20 \)
5. \( 3 \cdot 5 = 15 \)
6. \( 4 \cdot 4 = 16 \)

### Step 4: Substitute back into the equation
Now substituting these values back into the determinant expression:

\[ |U| = 2(24 - 25) - 3(18 - 20) + 4(15 - 16) \]

### Step 5: Simplify the terms
Calculating the differences:

1. \( 24 - 25 = -1 \)
2. \( 18 - 20 = -2 \)
3. \( 15 - 16 = -1 \)

Now substituting these back:

\[ |U| = 2(-1) - 3(-2) + 4(-1) \]

### Step 6: Calculate the final value
Calculating each term:

1. \( 2(-1) = -2 \)
2. \( -3(-2) = 6 \)
3. \( 4(-1) = -4 \)

Now combine these values:

\[ |U| = -2 + 6 - 4 = 0 \]

### Final Result
Thus, the value of \( |U| \) is

\[ \boxed{0} \]