Linear Algebra Practical Class, Week 22

Question One

Question

Find $A+B$ and $A\cdot B$, then answer whether or not the matrices commute, where

$$A=\left[\begin{array}{ccc}1& -1& 1\\ 0& 3& 1\\ 1& -2& 5\end{array}\right],B=\left[\begin{array}{ccc}1& 0& 0\\ -1& 2& 0\\ 3& -2& 3\end{array}\right]$$

Solution

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Question Two

Question

Find conditions for $a,b,c,d$ which make $A$: a diagonal matrix, a symmetric matrix, an upper triangle matrix, and a skew-symmetric matrix (such that $A^T=-A$), where

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$

Solution

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Question Three

Question

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

Solution

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Question Four

Question

Determine the rank of the following matrices, and of their matrix powers $A^2$ and $B^2$:

$$A=\left[\begin{array}{cc}2& -1\\ -1& \frac{1}{2}\end{array}\right]\text{and}B=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$

Solution

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Question Five

Question

Answer the following, where $A,B$ are any $n\times n$ matrices:

1. Is $(A-B)(A+B)=A^2-B^2$ true?

2. Show that $A{A}^T$ is a symmetric matrix.

Solution

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Question Six

Question

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

Solution

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