MTH1004 Linear Algebra Practical Class 7

Question 1

Question

Find the eigenvalues and eigenvectors of the following matrix: $B=\left(\begin{array}{ccc}1& -12& 4\end{array}\right)$. Show that its determinant equals to ${\lambda }_1{\lambda }_2$.

Solution

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Question 2

Question

Prove that any $2\times 2$ matrix of the form: $A=\left(\begin{array}{ccc}a& bb& a\end{array}\right)$ where $a\ne b$, $a$, $b$ in $\mathbb{R}$, has two real eigenvalues, which are: $a+b$, $a-b$.

Solution

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Question 3

Question

Diagonalise the matrix: $A=\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 1& 1& 0\end{array}\right)$.

Solution

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Question 4

Question

Find $B^n$ when $B=\left(\begin{array}{cc}0& 2\\ 1& 1\end{array}\right)$.

Solution

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Question 5

Question

Show that $A=\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)$ is non-diagonalisable. The matrix $B=\left(\begin{array}{cc}2& -1\\ 0& 2\end{array}\right)$ has two equal eigenvalues ${\lambda }_{1,2}=\lambda =2$. What is the algebraic and what is the geometric multiplicity of $\lambda$? Is $B$ diagonalisable?

Solution

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