Problem
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. \[ \begin{array}{r} 5 V_{1}+3 V_{2}-8 V_{3}=10 \\ 3 V_{1}-7 V_{2}+9 V_{3}=12 \\ -14 V_{1}+2 V_{2}+4 V_{3}=1 \end{array} \]
Solution
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): \(\boxed{(-8.967, -27.75, -17.261)}\)
