To solve this problem, we need to find the values of [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex] by equating the matrices on either side of the given equation. The steps below provide a detailed solution:
1. Calculate the right-hand side (RHS) of the equation:
[tex]\[
\text{RHS} = \frac{1}{2} \left( \begin{array}{cc}
16 & 8 \\
4 & -12
\end{array} \right)
\][/tex]
[tex]\[
\text{RHS} = \left( \begin{array}{cc}
\frac{1}{2} \times 16 & \frac{1}{2} \times 8 \\
\frac{1}{2} \times 4 & \frac{1}{2} \times -12
\end{array} \right)
\][/tex]
[tex]\[
\text{RHS} = \left( \begin{array}{cc}
8 & 4 \\
2 & -6
\end{array} \right)
\][/tex]
2. Calculate the left-hand side (LHS) of the equation:
- First, we calculate [tex]\(A\)[/tex]:
[tex]\[
A = 3 \left( \begin{array}{cc}
p & -1 \\
0 & 4
\end{array} \right)
\][/tex]
[tex]\[
A = \left( \begin{array}{cc}
3p & -3 \\
0 & 12
\end{array} \right)
\][/tex]
- Then, we subtract matrix [tex]\(B\)[/tex]:
[tex]\[
B = \left( \begin{array}{cc}
7 & q \\
-2 & 2r
\end{array} \right)
\][/tex]
- Finally, we compute [tex]\(A - B\)[/tex]:
[tex]\[
A - B = \left( \begin{array}{cc}
3p & -3 \\
0 & 12
\end{array} \right) - \left( \begin{array}{cc}
7 & q \\
-2 & 2r
\end{array} \right)
\][/tex]
[tex]\[
A - B = \left( \begin{array}{cc}
3p - 7 & -3 - q \\
0 - (-2) & 12 - 2r
\end{array} \right)
\][/tex]
[tex]\[
A - B = \left( \begin{array}{cc}
3p - 7 & -3 - q \\
2 & 12 - 2r
\end{array} \right)
\][/tex]
3. Equate the LHS and RHS matrices:
[tex]\[
\left( \begin{array}{cc}
3p - 7 & -3 - q \\
2 & 12 - 2r
\end{array} \right) = \left( \begin{array}{cc}
8 & 4 \\
2 & -6
\end{array} \right)
\][/tex]
By equating the corresponding elements, we obtain the following system of equations:
[tex]\[
3p - 7 = 8 \quad \cdots \quad (1)
\][/tex]
[tex]\[
-3 - q = 4 \quad \cdots \quad (2)
\][/tex]
[tex]\[
12 - 2r = -6 \quad \cdots \quad (3)
\][/tex]
4. Solve the system of equations for [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex]:
(i) For [tex]\( p \)[/tex]:
[tex]\[
3p - 7 = 8
\][/tex]
[tex]\[
3p = 15
\][/tex]
[tex]\[
p = 5
\][/tex]
[tex]\[
\boxed{5}
\][/tex]
(ii) For [tex]\( q \)[/tex]:
[tex]\[
-3 - q = 4
\][/tex]
[tex]\[
-q = 7
\][/tex]
[tex]\[
q = -7
\][/tex]
[tex]\[
\boxed{-7}
\][/tex]
(iii) For [tex]\( r \)[/tex]:
[tex]\[
12 - 2r = -6
\][/tex]
[tex]\[
12 + 6 = 2r
\][/tex]
[tex]\[
18 = 2r
\][/tex]
[tex]\[
r = 9
\][/tex]
[tex]\[
\boxed{9}
\][/tex]
Therefore, the values are:
(i) [tex]\( p = 5 \)[/tex]
(ii) [tex]\( q = -7 \)[/tex]
(iii) [tex]\( r = 9 \)[/tex]
