Linear Algebra Mid-Term Prediction

MTH1004 Linear Algebra Mid-Term Test

Duration: 1 Hour
Total Marks: 100
Note: A basic calculator for arithmetic may be used.

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Problem 1. (25 Marks)
Given the matrix

$B=\left(\begin{array}{ccc}2& -1& 0\\ 1& 2& -2\\ 1& 0& 1\end{array}\right)$

a) Find the eigenvalues and eigenvectors of matrix $B$.

b) Determine if $B$ is diagonalisable. If so, find a diagonal matrix $D$ such that $P^{-1}BP=D$, where $P$ is the matrix of eigenvectors.

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Problem 2. (25 Marks)
Consider the homogeneous system given by $C\mathbf{x}=0$, where

$C=\left(\begin{array}{ccc}1& 2& -1\\ -2& 4& 2\\ 3& 6& -3\end{array}\right)$

a) Solve the system using Gauss-Jordan elimination.

b) Show that the solution set forms a vector subspace of ${\mathbb{R}}^3$. Determine and justify a basis for this subspace.

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Problem 3. (25 Marks)
Let matrix $M$ be defined as

$M=\left(\begin{array}{ccc}0& 1& 2\\ 2& 0& -2\\ 1& -1& 1\end{array}\right)$

a) Using Gauss-Jordan elimination, find the inverse of $M$, if it exists.

b) If $N$ is a matrix such that $NM=I$, where $I$ is the identity matrix, show that $N=M^{-1}$.

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Problem 4. (25 Marks)
Given vectors $v_1=(1,2,3)$, $v_2=(4,5,6)$, and $v_3=(7,8,10)$ in ${\mathbb{R}}^3$,

a) Prove that $\{v_1,v_2,v_3\}$ forms a basis for ${\mathbb{R}}^3$.

b) …

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Attempt all problems. Each problem counts for 25 marks. Good luck!

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