To find the determinant of the product of the matrices [tex]\( P \)[/tex] and [tex]\( Q \)[/tex], follow these steps carefully:
1. Define the matrices:
First, we have the matrices [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] given as:
[tex]\[
P = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}, \quad
Q = \begin{pmatrix} 7 & 4 \\ -3 & 2 \end{pmatrix}
\][/tex]
2. Compute the product [tex]\( PQ \)[/tex]:
To find [tex]\( PQ \)[/tex], we perform the matrix multiplication of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[
PQ = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 7 & 4 \\ -3 & 2 \end{pmatrix}
\][/tex]
Carry out the multiplication element-by-element:
[tex]\[
(PQ)_{11} = 2 \cdot 7 + 1 \cdot (-3) = 14 - 3 = 11
\][/tex]
[tex]\[
(PQ)_{12} = 2 \cdot 4 + 1 \cdot 2 = 8 + 2 = 10
\][/tex]
[tex]\[
(PQ)_{21} = 3 \cdot 7 + 4 \cdot (-3) = 21 - 12 = 9
\][/tex]
[tex]\[
(PQ)_{22} = 3 \cdot 4 + 4 \cdot 2 = 12 + 8 = 20
\][/tex]
Therefore, the matrix [tex]\( PQ \)[/tex] is:
[tex]\[
PQ = \begin{pmatrix} 11 & 10 \\ 9 & 20 \end{pmatrix}
\][/tex]
3. Calculate the determinant of [tex]\( PQ \)[/tex]:
The determinant of a 2x2 matrix [tex]\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)[/tex] is given by [tex]\( ad - bc \)[/tex]. For the matrix [tex]\( PQ \)[/tex]:
[tex]\[
PQ = \begin{pmatrix} 11 & 10 \\ 9 & 20 \end{pmatrix}
\][/tex]
The determinant is:
[tex]\[
\text{det}(PQ) = (11 \cdot 20) - (10 \cdot 9)
\][/tex]
Performing the calculations:
[tex]\[
\text{det}(PQ) = 220 - 90 = 130
\][/tex]
Therefore, the determinant of [tex]\( PQ \)[/tex] is:
[tex]\[
\boxed{130}
\][/tex]
