Linear Algebra Mid-Term Prediction 2

Linear Algebra Exam - MTH1004

Question 1: Eigenvalues and Eigenvectors (25 marks)

Consider the matrix $A=\left[\begin{array}{cc}4& 2\\ 1& 3\end{array}\right]$:

1. Find the eigenvalues of matrix $A$. (10 marks)
2. For each eigenvalue, determine the corresponding eigenvectors. Show your work using the Gauss-Jordan Elimination process. (15 marks)

Question 2: Diagonalisation (25 marks)

Given a matrix $B=\left[\begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 5\end{array}\right]$:

1. Verify if matrix $B$ is diagonalisable. (5 marks)
2. Assuming $B$ is diagonalisable, find a matrix $P$ of eigenvectors and a diagonal matrix $D$ such that $P^{-1}BP=D$. Demonstrate the process explicitly. (20 marks)

Question 3: Solving Linear Systems and Matrix Inverse (25 marks)

Consider the system of equations given by:

$$\begin{array}{rl}x+2y-z& =4\\ 2x-y+3z& =-6\\ -x+3y+2z& =7\end{array}$$

1. Solve the system using Gauss-Jordan Elimination. (15 marks)
2. Determine if the coefficient matrix of the system is invertible. If it is, find the inverse using the Gauss-Jordan method. (10 marks)

Question 4: Basis, Dimension, and Cramer’s Rule (25 marks)

Consider the vectors $\mathbf{u}=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)$, $\mathbf{v}=\left(\begin{array}{c}4\\ 5\\ 6\end{array}\right)$, and $\mathbf{w}=\left(\begin{array}{c}7\\ 8\\ 9\end{array}\right)$ in ${\mathbb{R}}^3$:

1. Determine if the set $\{\mathbf{u},\mathbf{v},\mathbf{w}\}$ is a basis for ${\mathbb{R}}^3$. (10 marks)
2. Use Cramer’s Rule to solve the following system, if possible: $$\begin{array}{rl}x+4y+7z& =2\\ 2x+5y+8z& =3\\ 3x+6y+9z& =4\end{array}$$ Include the determinant calculations in your solution. (15 marks)