If `|(2,4,0),(0,5,16),(0,0,1+p)| = 20`, then what is the value of p ?

Options

A. 0
B. 1
C. 2
D. 5

Answer

To solve the problem, we need to find the value of \( p \) given the determinant equation:

\[ \left| \begin{array}{ccc} 2 & 4 & 0 \\ 0 & 5 & 16 \\ 0 & 0 & 1 + p \end{array} \right| = 20 \]

### Step 1: Calculate the Determinant
The determinant of a 3x3 matrix can be calculated using the formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:

\[ \begin{array}{ccc} a = 2 & b = 4 & c = 0 \\ d = 0 & e = 5 & f = 16 \\ g = 0 & h = 0 & i = 1 + p \end{array} \]

Using the determinant formula, we can simplify the calculation since \( c = 0 \):

\[ \text{det}(A) = 2 \left( 5(1 + p) - 16(0) \right) - 4 \left( 0(1 + p) - 0(16) \right) + 0 \]

This simplifies to:

\[ \text{det}(A) = 2 \cdot 5(1 + p) = 10(1 + p) \]

### Step 2: Set the Determinant Equal to 20
Now we set the determinant equal to 20:

\[ 10(1 + p) = 20 \]

### Step 3: Solve for \( p \)
To isolate \( p \), divide both sides by 10:

\[ 1 + p = 2 \]

Now, subtract 1 from both sides:

\[ p = 2 - 1 \]

Thus, we find:

\[ p = 1 \]

### Conclusion
The value of \( p \) is:

\[ \boxed{1} \]

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