MTH1004 Linear Algebra Tutorial Class 3

Question 1

Question

Problem 1: Consider the following matrices: $A_1=\left(\begin{array}{ccc}2& 3& -1\\ 0& 2& -1\end{array}\right)$, $A_2=\left(\begin{array}{ccc}1& 0& -1\\ 3& 0& -1\\ -1& 0& 4\end{array}\right)$, $A_3=\left(\begin{array}{ccc}1& 0& -2\\ 0& 1& 2\\ 0& 0& 0\end{array}\right)$, $A_4=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)$, $B_1=\left(\begin{array}{cccc}1& -1& -1& 0\\ 1& -1& 0& 0\\ 1& 0& -1& 0\\ 0& -1& 0& -1\end{array}\right)$, $B_2=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 1\end{array}\right)$, $B_3=\left(\begin{array}{ccc}1& 4& 0\\ 1& 0& 0\end{array}\right)$, $B_4=\left(\begin{array}{cccc}2& 0& 1& 3\\ -8& 0& 0& -1\\ -1& 3& 0& 0\\ 4& 7& 7& 0\end{array}\right)$. Find the matrices which are in row-echelon form and compute their rank. Which of those matrices are in reduced row-echelon form?

Solution

…

Question 2

Question

Problem 2: Show that: (i) $A{A}^T$ and (ii) $A+A^T$ are symmetric matrices.

Solution

…

Question 3

Question

Problem 3: Find $x$ when: $\left(\begin{array}{cccc}2& -1& 1& 3\end{array}\right)\left(\begin{array}{c}x\\ 3\\ 2\\ -1\end{array}\right)=\left(\begin{array}{c}-6\\ 7\\ 4\\ 0\end{array}\right)$.

Solution

…