 # Learn Matrices: Practice Problems and Examples

Recent questions in Matrices Maui1opj 2023-03-30

## Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix$\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$ parheliubdr 2023-02-15

## Find characteristic equation for the matrix $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ Rythalltiys 2023-02-13

## How do I calculate the determinant of a 4x4 Matrix? Mainzemj2w 2023-02-10

## What is the dimension of the matrix $\left[\begin{array}{c}7\\ 8\\ 9\end{array}\right]$? Tyson Haas 2023-02-07

## What is the dimension of the matrix $\left[\begin{array}{cc}16& 8\\ 10& 5\\ 0& 0\end{array}\right]$? casonehvc 2023-02-01

## How do you multiply 2x3 and 2x2 matrices? supermamanswtk 2023-01-07

## Which equation demonstrates the multiplicative identity property? A$\left(-3+5i\right)+0=-3+5i$ B$\left(-3+5i\right)\left(1\right)=-3+5i$ C$\left(-3+5i\right)\left(-3+5i\right)=-16-30i$ D$\left(-3+5i\right)\left(3-5i\right)=16+30i$

PrecalculusOpen question shatichome 2022-11-25

## Answer this equation ${x}^{2}-1=0$ Brandon White 2022-11-25

## How do I calculate the derivative of matrix?I'd like to expand a real, symmetric and positive definite Matrix $M$ into a Taylor series. I'm trying to do$M\left(T\right)=M\left({T}_{0}\right)+\frac{\mathrm{\partial }M}{\mathrm{\partial }T}\left(T-{T}_{0}\right)+\cdots$$T$ is a algebraic vector of parameters (e.g. temperatures at finite element nodes). I'm only interested in the first order term, i.e. the derivative of $M$ at ${T}_{0}$. My Professor tried to do this, but used unclear notation. Annie French 2022-11-17

## If we let R be the ring of $2×2$ complex matrices. When is the left annihilator just equal to {0}? I see that if A is invertible ${\text{Ann}}_{R}\left(A\right)$ is trivial since if $M\in {\text{Ann}}_{R}\left(A\right)$ then $MA=0$ so we can just multiply on the right by ${A}^{-1}$ and so $M=0$ Layton Park 2022-11-17

## What is the determinant of the following Householder matrix$I-2\stackrel{^}{x}{\stackrel{^}{x}}^{\mathrm{\top }}$where $\stackrel{^}{x}$ is a normalized vector, i.e., $‖\stackrel{^}{x}‖=1$?This is always −1 somehow but can't find proof of it anywhere. Laila Murphy 2022-11-16

## Let $A,B$ be square matrices. $A\ne 0$ (as a matrix) and ${A}^{2}=AB$. How can I prove that $|\left(A-B\right)|=0$? I think that key is a chain $|\left(A-B\right)|=\cdots =|\left(B-A\right)|$ but not sure if it's the right way. And prove that 0 is A-eigenvalue. Adison Rogers 2022-11-16

## Determine as the parameters $a,b\in \mathbb{R}$ the rank of the following matrix$A=\left(\begin{array}{ccc}a+b& a& b\\ b& a& 0\\ a& b& b\end{array}\right)$ Keshawn Moran 2022-11-13

## Suppose I have matrix A and vectors B and C. I assume two facts about these matrices:(1) Assume that $AB=AC$ for rectangular matrix $A\in {\mathbb{R}}^{a×b}$ with rank r where $0(2) Assume that $\left({I}_{b}-{A}^{\prime }\left(A{A}^{\prime }{\right)}^{-}A\right)C=0$ where $\left(A{A}^{\prime }{\right)}^{-}$ is the pseudo-inverse of $A{A}^{\prime }$Do (1) and (2) together imply that $\left(I-{A}^{\prime }\left({A}^{\prime }A{\right)}^{-}A\right)B=0$? Elliana Molina 2022-11-13

## Let $X,A\in {\mathbb{C}}^{n×n}$ and suppose$0\le X\le \text{Id},\phantom{\rule{1em}{0ex}}0\le A,$where $\text{Id}\in {\mathbb{C}}^{n×n}$ denotes the identity matrix. Is it true that$XAX\le A,$or can you give a counterexample? kemecryncqe9 2022-11-11

## Find the value of the following determinant.$det\left(\begin{array}{ccc}{x}^{2}& \left(x+1{\right)}^{2}& \left(x+2{\right)}^{2}\\ \phantom{\rule{mediummathspace}{0ex}}{y}^{2}& \left(y+1{\right)}^{2}& \left(y+2{\right)}^{2}\\ \phantom{\rule{mediummathspace}{0ex}}{z}^{2}& \left(z+1{\right)}^{2}& \left(z+2{\right)}^{2}\end{array}\right)$ assupecoitteem81 2022-11-11

## Let $x,y,z$ are three $n×1$ vectors. For each vector, every element is between 0 and 1, and the sum of all elements in each vector is 1. Why the following inquality holds:${x}^{T}y+{y}^{T}z-{x}^{T}z\le 1$ Kenna Stanton 2022-11-10

## If ${A}^{2}=I,{B}^{2}=\left[\begin{array}{cc}3& 2\\ -2& -1\end{array}\right],AB=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, find BA.What I did was the following: ${A}^{2}=I;A{A}^{-1}=I\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}{A}^{2}=A{A}^{-1}$$\left(AB{\right)}^{2}=ABAB\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}{\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]}^{2}=I\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}ABAB=I\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}BA={A}^{-1}{B}^{-1}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}BA=\left(BA{\right)}^{-1}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}BA=\left[\begin{array}{cc}±1& 0\\ 0& ±1\end{array}\right]$which I'm not sure whether it's correct or not and if it's correct, which of these ±s to select from. Paula Cameron 2022-11-07
## Multiplying Matrix X by ${X}^{T}=2{X}^{T}$?In Elements of Statistical Learning, we differentiate $RSS\left(\beta \right)=\left(y-X\beta {\right)}^{T}\left(y-X\beta \right)$ w.r.t to $\beta$ to get ${X}^{T}\left(y-X\beta \right)$According to some, this is because$\left(y-X\beta {\right)}^{T}\left(y-X\beta \right)={y}^{T}y-2{\beta }^{T}{X}^{T}y+{\beta }^{T}{X}^{T}X\beta$ MISA6zh 2022-11-04