a) Evaluate the following, expressing your answers in whole numbers:
$[\begin{array}{ll} 1 & 4 \\ 9 & 6 \end{array}] \times [\begin{matrix} 7 \\ 10 \end{matrix}] [\begin{array}{cc} 22 & 34 \\ 11 & 12 \end{array}] \times [\begin{array}{cc} 76 & 62 \\ 38 & - 44 \end{array}]$
b) Calculate the determinant of the following two matrices, expressing your answers in whole numbers:
$[\begin{array}{ccc} 3 & 4 & - 10 \\ 12 & - 4 & 10 \\ 15 & 6 & 6 \end{array}] [\begin{array}{ccc} 67 & - 12 & 22 \\ 78 & - 65 & 90 \\ - 55 & 7 & 43 \end{array}]$
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

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Answer

Node voltages (V1, V2, V3): $(- 8.967, - 27.75, - 17.261)$

Steps

Step 1 ：Matrix AB: $[\begin{matrix} 47 \\ 123 \end{matrix}]$

Step 2 ：Matrix CD: $[\begin{matrix} 2964 & - 132 \\ 1292 & 154 \end{matrix}]$

Step 3 ：Determinant of E: $- 1260$

Step 4 ：Determinant of F: $- 196464$

Step 5 ：x = $1.463$

Step 6 ：y = $- 0.659$

Step 7 ：Node voltages (V1, V2, V3): $(- 8.967, - 27.75, - 17.261)$