Linear Algebra Tutorial Class, Week 23

Question One

Question

1. If $A$ is invertible, prove that a homogenous system $A\mathbf{x}=0$ has a unique solution: the zero solution $\mathbf{x}=0$.
2. If $A$ is invertible, what is the rank of $A$? Justify your answer.
3. Simplify the expression $(A^{-1}{B}^2{C}^{-2}{)}^{-1}$.
4. Prove that if $A$ satisfies the matrix equation $4A^3-A^2-2I=O$, then $A$ is invertible.
5. Solve the matrix equation $A^{-1}X+B+3A=O$ in terms of the matrix $X$.

Solution

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Question Two

Question

Find the inverses of the following matrices, where any $2\times 2$ matrices are completed with the standard formula and any $3\times 3$ are with Gauss-Jordan elimination:

1. $A=\left[\begin{array}{cc}1& -2\\ 5& 7\end{array}\right]$
2. $R=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$
3. $C=\left[\begin{array}{ccc}1& 1& 2\\ 1& 2& 1\\ 2& 1& 1\end{array}\right]$

By then using the inverse $C^{-1}$, solve the system $C\mathbf{x}=\mathbf{u}$, where $\mathbf{u}=\left[\begin{array}{c}2\\ 1\\ 0\end{array}\right]$.

Solution

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