Linear Algebra Tutorial Class, Week 22

Question One

Question

Consider the following matrices, find which are in row-echelon form and compute their rank, then, find which of those matrices are in reduced row-echelon form:

$$A_1=\left[\begin{array}{ccc}2& 3& -1\\ 0& 2& -1\end{array}\right],A_2=\left[\begin{array}{cccc}1& 0& -1& 3\\ 0& -1& -1& 0\\ 0& 0& 0& 4\end{array}\right],A_3=\left[\begin{array}{ccc}1& 0& -2\\ 0& 1& 2\\ 0& 0& 0\end{array}\right],A_4=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right],$$ $$B_1=\left[\begin{array}{ccc}1& -1& -1\\ 0& 1& -1\\ 0& 0& 1\\ 0& 0& -1\end{array}\right],B_2=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],B_3=\left[\begin{array}{cc}1& 4\\ 0& 1\\ 0& 0\end{array}\right],B_4=\left[\begin{array}{ccccc}2& 0& 1& 3& -8\\ 0& 0& -1& -1& 3\\ 0& 0& 4& 7& 7\end{array}\right].$$

Solution

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Question Two

Question

Show that both $A{A}^T$ and $A+A^T$ are symmetric matrices.

Solution

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Question Three

Question

Find $x$ when:

$$\left[\begin{array}{cc}2& -1\\ 1& 3\end{array}\right]\left[\begin{array}{cc}x& 3\\ 2& -1\end{array}\right]=\left[\begin{array}{cc}-6& 7\\ 4& 0\end{array}\right]$$

Solution

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