Question

Let A, B,C be following matrices:

a) Calculate the inverse matrices A- and B¬l if exist. b) Calculate ranks of A, B and C. \text { c) Calculate a basis of } \operatorname{Ker}\left(f_{B}\right):=\left\{\mathbf{x} \in \mathbb{R}^{3} \mid f_{B}(\mathbf{x})=\mathbf{0}\right\} \text { of the following } \mathbb{R} \text {-linear map: } f_{B}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} ; \quad \mathbf{x}:=\left(\begin{array}{l}

x_{1} \\

x_{2} \\

x_{3}

\end{array}\right) \mapsto f_{B}(\mathbf{x}):=\left(\begin{array}{ccc}

1 & 2 & 3 \\

4 & 5 & 6 \\

5 & 7 & 9

\end{array}\right)\left(\begin{array}{l}

x_{1} \\

x_{2} \\

x_{3}

\end{array}\right) \text { Calculate a basis of } \operatorname{Im}\left(f_{B}\right):=\left\{\mathbf{y} \in \mathbb{R}^{3} \mid \exists \mathbf{x} \in \mathbb{R}^{3} \text { such that } \mathbf{y}=f_{B}(\mathbf{x})\right\}

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