MTH1004 Linear Algebra Practical Class 4

Question 1

Question

Find $A+B$ and $A\cdot B$, when $A=\left[\begin{array}{ccc}1& -1& 1\\ 0& 3& 1\\ 1& -2& 5\end{array}\right]$, $B=\left[\begin{array}{ccc}1& 0& 0\\ -1& 2& 0\\ 3& -2& 3\end{array}\right]$. Do the matrices A, B commute ($AB=BA$)?

Solution

…

Question 2

Question

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$. Find conditions for $a,b,c,d$ which make A: (a) A diagonal matrix (b) A symmetric matrix (c) An upper triangular matrix (d) A skew-symmetric matrix ($A^T=-A$).

Solution

…

Question 3

Question

Let $Ax=0$ be a homogeneous system with 3 equations and 3 unknowns. Find the rank of the matrix A, when the system has: (a) a unique solution. (b) infinite solutions of the form $x=2z,y=-z,z$ in $\mathbb{R}$.

Solution

…

Question 4

Question

Determine the rank of following matrices $A=\left[\begin{array}{cc}2& -1\\ -1& 1/2\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ and of their matrix powers $A^2$ and $B^2$. (a) Is $(A-B)(A+B)=A^2-B^2$ true for any $n\times n$ A, B matrices? Justify your answer. (b) Show that $A{A}^T$ is a symmetric matrix.

Solution

…

Question 5

Question

Let $u,v$ be two linearly independent vectors in ${\mathbb{R}}^2$ and $A$ the matrix with columns the vectors $u,v$. What is the rank of A?

Solution

…