To solve this problem, we need to find the value of $p$ so that the determinants of the two given matrices are equal.

Step 1: Write down the matrices

The two matrices are:

1. $\begin{pmatrix} 2 & -4 \\ p & 5 \end{pmatrix}$
2. $\begin{pmatrix} -3 & 2 \\ -2 & -6 \end{pmatrix}$

Step 2: Calculate the determinants of both matrices

For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is calculated as: $\text{determinant} = a \cdot d - b \cdot c$

Determinant of the first matrix

$\det\begin{pmatrix} 2 & -4 \\ p & 5 \end{pmatrix} = 2 \cdot 5 - (-4) \cdot p = 10 + 4p$

Determinant of the second matrix

$\det\begin{pmatrix} -3 & 2 \\ -2 & -6 \end{pmatrix} = (-3) \cdot (-6) - (2) \cdot (-2) = 18 + 4 = 22$

Step 3: Set the determinants equal to each other

Since the determinants are given to be equal: $10 + 4p = 22$

Step 4: Solve for $p$

Subtract 10 from both sides: $4p = 12$ Now, divide by 4: $p = 3$

Answer

The value of $p$ is $\boxed{3}$, so the correct answer is D. 3.

Would you like more details or have any questions?

Related Questions

1. How do you calculate the determinant of a 3x3 matrix?
2. What happens if two matrices have the same determinant but different entries?
3. Can a matrix with a determinant of zero be inverted?
4. How can you use determinants to check if two vectors are linearly dependent?
5. What is the geometric interpretation of the determinant in 2D space?

Tip

For any $2 \times 2$ matrix, always remember the formula $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$. It is a quick way to calculate the determinant!