a) Evaluate the following, expressing your answers in whole numbers:
\[
\left[\begin{array}{ll}
1 & 4 \\
9 & 6
\end{array}\right] \times\left[\begin{array}{c}
7 \\
10
\end{array}\right] \quad\left[\begin{array}{cc}
22 & 34 \\
11 & 12
\end{array}\right] \times\left[\begin{array}{cc}
76 & 62 \\
38 & -44
\end{array}\right]
\]
b) Calculate the determinant of the following two matrices, expressing your answers in whole numbers:
\[
\left[\begin{array}{ccc}
3 & 4 & -10 \\
12 & -4 & 10 \\
15 & 6 & 6
\end{array}\right] \quad\left[\begin{array}{ccc}
67 & -12 & 22 \\
78 & -65 & 90 \\
-55 & 7 & 43
\end{array}\right]
\]
c) Use the Gaussian elimination method to solve the following set of simultaneous equations, expressing your answers to three significant figures:
\[
\begin{array}{l}
5 x+2 y=6 \\
3 x-7 y=9
\end{array}
\]
d) The node voltages of a simple, dc resistor circuit are given by the following set of simultaneous equations. Evaluate the node voltages using Cramer's rule, expressing your answers to 3 significant figures.
\[
\begin{array}{r}
5 V_{1}+3 V_{2}-8 V_{3}=10 \\
3 V_{1}-7 V_{2}+9 V_{3}=12 \\
-14 V_{1}+2 V_{2}+4 V_{3}=1
\end{array}
\]
Answer
Node voltages (V1, V2, V3): \(\boxed{(-8.967, -27.75, -17.261)}\)
Step 1 ：Matrix AB: \(\begin{bmatrix} 47 \\ 123 \end{bmatrix}\)
Step 2 ：Matrix CD: \(\begin{bmatrix} 2964 & -132 \\ 1292 & 154 \end{bmatrix}\)
Step 3 ：Determinant of E: \(\boxed{-1260}\)
Step 4 ：Determinant of F: \(\boxed{-196464}\)
Step 5 ：x = \(\boxed{1.463}\)
Step 6 ：y = \(\boxed{-0.659}\)
Step 7 ：Node voltages (V1, V2, V3): \(\boxed{(-8.967, -27.75, -17.261)}\)
