Recent questions in Matrices

PrecalculusAnswered question

Maui1opj 2023-03-30

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix

$$\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$$

PrecalculusAnswered question

parheliubdr 2023-02-15

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

PrecalculusAnswered question

Rythalltiys 2023-02-13

How do I calculate the determinant of a 4x4 Matrix?

PrecalculusAnswered question

Mainzemj2w 2023-02-10

What is the dimension of the matrix $\left[\begin{array}{c}7\\ 8\\ 9\end{array}\right]$?

PrecalculusAnswered question

Tyson Haas 2023-02-07

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

PrecalculusAnswered question

supermamanswtk 2023-01-07

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

PrecalculusAnswered question

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(T)=M({T}_{0})+\frac{\mathrm{\partial}M}{\mathrm{\partial}T}(T-{T}_{0})+\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.

I'd like to expand a real, symmetric and positive definite Matrix $M$ into a Taylor series. I'm trying to do

$M(T)=M({T}_{0})+\frac{\mathrm{\partial}M}{\mathrm{\partial}T}(T-{T}_{0})+\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.

PrecalculusAnswered question

Annie French 2022-11-17

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

PrecalculusAnswered question

Layton Park 2022-11-17

What is the determinant of the following Householder matrix

$I-2\hat{x}{\hat{x}}^{\mathrm{\top}}$

where $\hat{x}$ is a normalized vector, i.e., $\Vert \hat{x}\Vert =1$?

This is always −1 somehow but can't find proof of it anywhere.

$I-2\hat{x}{\hat{x}}^{\mathrm{\top}}$

where $\hat{x}$ is a normalized vector, i.e., $\Vert \hat{x}\Vert =1$?

This is always −1 somehow but can't find proof of it anywhere.

PrecalculusAnswered question

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 $|(A-B)|=0$? I think that key is a chain $|(A-B)|=\cdots =|(B-A)|$ but not sure if it's the right way. And prove that 0 is A-eigenvalue.

PrecalculusAnswered question

Adison Rogers 2022-11-16

Determine as the parameters $a,b\in \mathbb{R}$ the rank of the following matrix

$A=\left(\begin{array}{ccc}a+b& a& b\\ b& a& 0\\ a& b& b\end{array}\right)$

$A=\left(\begin{array}{ccc}a+b& a& b\\ b& a& 0\\ a& b& b\end{array}\right)$

PrecalculusAnswered question

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 $A\in {\mathbb{R}}^{a\times b}$ with rank r where $0<r<min(a,b)$

(2) Assume that $({I}_{b}-{A}^{\prime}(A{A}^{\prime}{)}^{-}A)C=0$ where $(A{A}^{\prime}{)}^{-}$ is the pseudo-inverse of $A{A}^{\prime}$

Do (1) and (2) together imply that $(I-{A}^{\prime}({A}^{\prime}A{)}^{-}A)B=0$?

(1) Assume that $AB=AC$ for rectangular matrix $A\in {\mathbb{R}}^{a\times b}$ with rank r where $0<r<min(a,b)$

(2) Assume that $({I}_{b}-{A}^{\prime}(A{A}^{\prime}{)}^{-}A)C=0$ where $(A{A}^{\prime}{)}^{-}$ is the pseudo-inverse of $A{A}^{\prime}$

Do (1) and (2) together imply that $(I-{A}^{\prime}({A}^{\prime}A{)}^{-}A)B=0$?

PrecalculusAnswered question

Elliana Molina 2022-11-13

Let $X,A\in {\mathbb{C}}^{n\times n}$ and suppose

$0\le X\le \text{Id},\phantom{\rule{1em}{0ex}}0\le A,$

where $\text{Id}\in {\mathbb{C}}^{n\times n}$ denotes the identity matrix. Is it true that

$XAX\le A,$

or can you give a counterexample?

$0\le X\le \text{Id},\phantom{\rule{1em}{0ex}}0\le A,$

where $\text{Id}\in {\mathbb{C}}^{n\times n}$ denotes the identity matrix. Is it true that

$XAX\le A,$

or can you give a counterexample?

PrecalculusAnswered question

kemecryncqe9 2022-11-11

Find the value of the following determinant.

$det\left(\begin{array}{ccc}{x}^{2}& (x+1{)}^{2}& (x+2{)}^{2}\\ \phantom{\rule{mediummathspace}{0ex}}{y}^{2}& (y+1{)}^{2}& (y+2{)}^{2}\\ \phantom{\rule{mediummathspace}{0ex}}{z}^{2}& (z+1{)}^{2}& (z+2{)}^{2}\end{array}\right)$

$det\left(\begin{array}{ccc}{x}^{2}& (x+1{)}^{2}& (x+2{)}^{2}\\ \phantom{\rule{mediummathspace}{0ex}}{y}^{2}& (y+1{)}^{2}& (y+2{)}^{2}\\ \phantom{\rule{mediummathspace}{0ex}}{z}^{2}& (z+1{)}^{2}& (z+2{)}^{2}\end{array}\right)$

PrecalculusAnswered question

assupecoitteem81 2022-11-11

Let $x,y,z$ are three $n\times 1$ 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:

${x}^{T}y+{y}^{T}z-{x}^{T}z\le 1$

${x}^{T}y+{y}^{T}z-{x}^{T}z\le 1$

PrecalculusAnswered question

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: ${A}^{2}=I;A{A}^{-1}=I\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{A}^{2}=A{A}^{-1}$

$(AB{)}^{2}=ABAB\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]}^{2}=I\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}ABAB=I\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}BA={A}^{-1}{B}^{-1}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}BA=(BA{)}^{-1}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}BA=\left[\begin{array}{cc}\pm 1& 0\\ 0& \pm 1\end{array}\right]$

which I'm not sure whether it's correct or not and if it's correct, which of these ±s to select from.

What I did was the following: ${A}^{2}=I;A{A}^{-1}=I\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{A}^{2}=A{A}^{-1}$

$(AB{)}^{2}=ABAB\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]}^{2}=I\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}ABAB=I\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}BA={A}^{-1}{B}^{-1}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}BA=(BA{)}^{-1}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}BA=\left[\begin{array}{cc}\pm 1& 0\\ 0& \pm 1\end{array}\right]$

which I'm not sure whether it's correct or not and if it's correct, which of these ±s to select from.

PrecalculusAnswered question

Paula Cameron 2022-11-07

Multiplying Matrix X by ${X}^{T}=2{X}^{T}$?

In Elements of Statistical Learning, we differentiate $RSS(\beta )=(y-X\beta {)}^{T}(y-X\beta )$ w.r.t to $\beta $ to get ${X}^{T}(y-X\beta )$

According to some, this is because

$$(y-X\beta {)}^{T}(y-X\beta )={y}^{T}y-2{\beta}^{T}{X}^{T}y+{\beta}^{T}{X}^{T}X\beta $$

In Elements of Statistical Learning, we differentiate $RSS(\beta )=(y-X\beta {)}^{T}(y-X\beta )$ w.r.t to $\beta $ to get ${X}^{T}(y-X\beta )$

According to some, this is because

$$(y-X\beta {)}^{T}(y-X\beta )={y}^{T}y-2{\beta}^{T}{X}^{T}y+{\beta}^{T}{X}^{T}X\beta $$

PrecalculusAnswered question

MISA6zh 2022-11-04

Suppose that M is an $n\times n$ matrix with entries given by

$${M}_{ij}={\rho}^{|i-j|}$$

for some $\rho \in (0,1)$. Is it true that the matrix M is non-negative definite (or even positive definite)?

I was trying to write M as ${A}^{\mathrm{\top}}A$ for some matrix A, but could not do so. Any help would be greatly appreciated.

$${M}_{ij}={\rho}^{|i-j|}$$

for some $\rho \in (0,1)$. Is it true that the matrix M is non-negative definite (or even positive definite)?

I was trying to write M as ${A}^{\mathrm{\top}}A$ for some matrix A, but could not do so. Any help would be greatly appreciated.

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