Tutorial Week 7
Cramer’s Rule
Q1: Use Cramer’s Rule to solve for \(z\) in the system \(\begin{cases} 2x + 3y + z = 1 \\ x - 4y + z = -5 \\ 6x + y - 3z = 4 \end{cases}\).
Q2: Given \(AX = B\) where \(A = \begin{bmatrix} \Box & 2 & 1 \\ \Box & 5 & 3 \\ \Box & \Box & \Box \end{bmatrix}\), \(X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}\), and \(B = \begin{bmatrix} 0 \\ 0 \\ -3 \end{bmatrix}\), if \(det(A) = 5\), find \(x\).
Domains of multivariable functions
Q3: Find and sketch the domain of \(f(x, y) = \sqrt{2x^2 + 2y^2 - 8} - ln(x - y)\).
Q4: Find and sketch the domain of \(f(x, y) = \sqrt{3x + y} + 2^x - log_2(x^2-4)\).
Level sets
Q5: Sketch level sets of \(f(x, y) = 5x^2 + y^2\) for \(L(1)\), \(L(5)\), and \(L(9)\).
