This website uses cookies to improve your experience while you navigate through the website. Applications of maximal surfaces in Lorentz spaces. The answer has been given as: ( m 3 n n m + 2 n) I don't understand how they're getting that form. Unexpected low characteristic impedance using the JLCPCB impedance calculator. 3) When i solve the equation $AB=BA$, I get this: $$\left[\begin{array}{cc}ae+bg&af+bh\\ce+dg&cf+dh\end{array}\right]=\left[\begin{array}{cc}ea+cf&eb+df\\ga+ch&gb+dh\end{array}\right]$$. Which comes first: CI/CD or microservices. the matrix of the form $$\pmatrix{a&0\\0&a}$$ called the scalar matrix is the matrix that commutes with all $2 \times 2$ matrices. Furthermore you probably know that all multiples of the identity matrix commute with $A$. Learn more about Stack Overflow the company, and our products. Set this to zero and solve for . Eigenvector: An eigenvector of A is a nonzero vector v in Rn such that Av=v, for some scalar . Cookie Notice Considering we have to multiply entry 1-2 with entry 2-1, this would mean we're mulitplying the same value if the matrix is symmetric, i.e. A B = [ a b c d] [ p q r s] = [ a p + b r a q + b s c p + d r c q + d s] Find BA. This answer is correct, but could you show me how to prove that result is (k,0)(0,k). Determine all 2 2 matrices A such that A2 = 0. The last equation, $\mathbf{B}=\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, give us $\mathbf{AB}=\mathbf{BA}$ when $b=0$, $c=0$, $a=d$. mean? So, the matrix is a - 2 - 2 a 3 3 - 3 - 2 c 3 Ia, c for which 2 3 is an eigenvector with associated eigenvalue -1. This problem has been solved! Why is static-static diffie hellman needed in Noise_IK? 94% of StudySmarter users get better grades. See Answer See Answer See Answer done loading. linear Alegbra . Igor Konovalov. Playing a game as it's downloading, how do they do it? By rejecting non-essential cookies, Reddit may still use certain cookies to ensure the proper functionality of our platform. This question has confused me a bit. squaring it. Solutions: (1 0 0 1), ( 1 0 0 1), (1 0 c 1), ( 1 0 c 1),( a b 1a2 b a) Explanation: Suppose A = ( a b c d) Then: A2 = ( a b c d)(a b c d) = ( a2 + bc b(a + d) c(a + d) d2 + bc) So if we want A2 = (1 0 0 1) then we get the following equations: a2 +bc = 1 b(a +d) = 0 c(a +d) = 0 d2 +bc = 1 Draw a sketch. Advanced Math Advanced Math questions and answers Let a) Find all matrices such that AB = BA.b) Use the results of part a to show (2x2) matrices B and C such that AB=BA and AC This problem has been solved! Korbanot only at Beis Hamikdash ? So if entry 1-1 is a, the first multiplication is a*a = a2. squaring it. So in this case P is equal to (-5) (+1). You will find that the characterisic equation is quadratic and can have non-real roots even if coefficients are real. If v is an Eigen vector of A this means that: Av=vabcd23=-1232a+3b2c+3d=-2-3By the equality of matrices,2a+3b=-22c+3d=-3. Sign up for free to discover our expert answers. It only takes a minute to sign up. Question: Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10. Complexity of |a| < |b| for ordinal notations? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. See Answer Find all 2x2 matrices for which [23] is an eigenvector with associated eigenvalue -1. You can try to see if that is all. This category only includes cookies that ensures basic functionalities and security features of the website. I don't think there is one other than the zero matrix itself. So if entry 1-1 is a, the first multiplication is a*a = a 2. what are the $2 \times 2$ matrices $\pmatrix{a&b\\c&d}$ that commute with $\pmatrix{0&1\\0&0}?$, $$ \pmatrix{0&a\\0&c} = \pmatrix{a&b\\c&d} \pmatrix{0&1\\0&0} = \pmatrix{0&1\\0&0}\pmatrix{a&b\\c&d}=\pmatrix{c&d\\0&0}$$ implies we need $$ a= d, c = 0$$, by considering the matrix $\pmatrix{0&0\\1&0}$ you will find $$ a= d, b = 0$$. Sort by: Top Voted Jonathan Nativ 11 years ago What is the use of finding the determinant of a matrix? However, sometimes it's hard to find all the solutions of a set of equations, and/or hard to describe them elegantly. We can then try $\mathbf{B}=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$. We also use third-party cookies that help us analyze and understand how you use this website. Is it possible for rockets to exist in a world that is only in the early stages of developing jet aircraft? We'll assume you're ok with this, but you can opt-out if you wish. Is it bigamy to marry someone to whom you are already married? If you just need one example, the fastest way to find one is to use trial and error. For which angle(s) can you find three distinct unit vectors in 2 such that the angle between any two of them is ? Why is Bb8 better than Bc7 in this position? Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Find all symmetric 2x2 matrices A such that A^2 = 0 That's the question. Reddit, Inc. 2023. Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10. This is exactly what you want: you want the $2\times 2$ matrices $z$ for which, for any $2\times 2$ matrix $g$, you have $zg=gz$. Necessary cookies are absolutely essential for the website to function properly. I started solving problem in this way: 1) I have this matrix A with real numbers: A =[a c b d] It is mandatory to procure user consent prior to running these cookies on your website. Let A = (a c b d) A2 = (a2 + bc ac + cd ab + bd d2 + cb) Equating all the four terms of the matrix to zero and solving, a2 + bc = 0 and d2 + bc = 0 gives a = d Solving ab + bd = 0 and ac + cd = 0 gives a = d OR b = c, which gives a = d = 0 Does a knockout punch always carry the risk of killing the receiver? $$A=\left[\begin{array}{cc}a &b\\c &d\end{array}\right]$$ I've been just trying to find two 2x2 matrices who result in a 2x2 matrix with 0 at each position. If u,v,w are three unit vectors in 3, what are the possible values of det[uvw]? $$Z(G) = \left\{ \left(\begin{array}{cc} k & 0 \\ 0 & k \end{array}\right) : k \in \mathbb{R}, \ k \neq 0 \right\}$$. AB=BA For any group $G$ the centre of $G$, denoted $Z(G)$, is defined as Created by Sal Khan. This website uses cookies to improve your experience. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. @USB_MAT I have edited my post to answer your question. See the step by step solution Step by Step Solution TABLE OF CONTENTS Step 1: Define the eigenvector This is problem from one book. The next value to be added to this must add up to -a2. In the case $n=2$ We can then solve for $\mathbf{B}=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ which gives us $\mathbf{AB}=\mathbf{BA}$ when $b=0$ and $c=0$. Don't have to recite korbanot at mincha? All matrices which commute with all 2 2 matrices Ask Question Asked 8 years, 4 months ago Modified 2 years, 8 months ago Viewed 8k times 3 I would like to find all matrices which commute with all 2 2 matrices. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Therefore, the required matrices is a-2-2a3c-3-2a3Ia,c. To find the eigenvalues you have to find a characteristic polynomial P which you then have to set equal to zero. We use the letter $Z$ because "zentrum" is "centre" in German. I only have answer (k 0 )(0 k) but I don't know how to get that result. ( 160 votes) Flag vaibhav singhal 11 years ago it helps in solving the system of linear equations 3 comments Try plugging in specific values for $e,f,g,h$ and see what equations come out (hint: use zeros and ones). @USB - for the sake of argument, how do you know that "this answer is correct"? Find the determinants of the linear transformations in Exercises 17 through 28. I would like to find all matrices which commute with all $2\times2$ matrices. Is there a reason beyond protection from potential corruption to restrict a minister's ability to personally relieve and appoint civil servants? Solving for $a,b,c$ & $d$ we get $\mathbf{AB}=\mathbf{BA}$ when $c=0$ and $b=0$. The proof can be found by Googling "Centre of the General Linear Group". The group of $n\times n$ matrices with real entries is called the general linear group and is denoted by $\mathrm{GL}(n,\mathbb{R})$. The identity matrix is the identity element of the group. Your notation is obscuring the fact that this equation has to hold for all possible $e,f,g,h$ using the same values of $a,b,c,d$. 10 years ago. Step 5 Find AB. What about for a 3x3 matrix, or an arbitrary nxn matrix? Privacy Policy. 2) Matrix which commute with matrix $A$ is matrix $B$: Answer & Explanation hesgidiauE Skilled 2021-03-05 Added 106 answers Step 1 Let A = [ a b c d] and B = [ p q r s] be any 2 2 matrices. You also have the option to opt-out of these cookies. Why does a rope attached to a block move when pulled? These cookies will be stored in your browser only with your consent. 1 comment. So, the matrix is a-2-2a33-3-2c3Ia,c for which 23 is an eigenvector with associated eigenvalue -1. Living room light switches do not work during warm/hot weather. For $\mathbf{B}=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ we get $\mathbf{AB}=\mathbf{BA}$ when $c=0$ and when $a=d$. I'm guessing the question just asked you to find an example? we can try four different scenarios with matrices that have 3 zero entires. Solve the equations 2a+3b=-2 and 2c+3d=-3d to get b=-2-2a3 and d=-3-2c3. How do the prone condition and AC against ranged attacks interact? By accepting all cookies, you agree to our use of cookies to deliver and maintain our services and site, improve the quality of Reddit, personalize Reddit content and advertising, and measure the effectiveness of advertising. How I can get the general look of wanted matrix? I don't think there is one other than the zero matrix itself. Find all 2x2 matrices A such that A^2 = I2. In your case, you have $2\times 2$ matrices, and so $n=2$. I started solving problem in this way: 1) I have this matrix $A$ with real numbers: A real-valued matrix can have complex-valued eigenvalues. How to make the pixel values of the DEM correspond to the actual heights? You'll get a detailed solution from a subject matter expert that helps you learn core concepts. It turns out that the centre of $\mathrm{GL}(n,\mathbb{R})$, for all $n$, is given by the matrices which are a multiple of the identity matrix. These cookies do not store any personal information. Connect and share knowledge within a single location that is structured and easy to search. For more information, please see our You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Ways to find a safe route on flooded roads. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. let A^2 = I2. Short Answer Find all 2x2 matrices for which [ 2 3] is an eigenvector with associated eigenvalue -1. Sal shows how to find the determinant of a 2x2 matrix. Find all symmetric 2x2 matrices A such that A^2 = 0. The set of $2\times 2$ matrices with non-zero determinant form a group under the operation of matrix multiplication. In particular if the characteristic equation were 2 + 1 = 0 2 + 1 = 0 you would have roots i i with i2 = 1 i 2 = 1. That's the question. Find all 2 2 matrices that commute with ( 2 3 1 4) My progress: I know that a square matrix commutes with itself, the identity matrix of that order, the null matrix of that order and any scalar matrix of that order. If you don't want a group structure then you can include $k=0$. So am I right in saying that no symmetric matrix A exists (other than the zero matrix, of course) such that A2 = 0? Find all $2\times 2$ matrices that commute with $AX = XA$? 1 Reply skullturf 7 yr. ago That's a noble goal. $$B = \left[\begin{array}{cc}e& f\\ g &h\end{array}\right]$$ But no such real value exists whose square will be negative. $\mathbf{A}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\mathbf{B}=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$, $\mathbf{B}=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $\mathbf{B}=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, $\mathbf{B}=\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, $\mathbf{A}=\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}$, All matrices which commute with all $2\times 2$ matrices, CEO Update: Paving the road forward with AI and community at the center, Building a safer community: Announcing our new Code of Conduct, AI/ML Tool examples part 3 - Title-Drafting Assistant, We are graduating the updated button styling for vote arrows, Find all $2 \times 2$ matrices $A$ such that $AB = BA$ for every $2 \times 2$ matrix $B$, Finding all elements in $GL_{2}(F)$ that commute with all other elements in $GL_{2}(F)$, Find all matrices that commute with given matrix, Which are matrices 22 that commute with the matrix, Find the matrices which commute with another, Find all matrices which commute with a given matrix, Find the values of the parameters for which the matrices commute. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. The problem statement is: Find a 2 x 2 matrix A such that A 2 = = -I The solution is available in the answer section of the book, but it is not shown how the author comes up with the solution. These equations imply that $a=d$ and $b=c=0$. The best answers are voted up and rise to the top, Not the answer you're looking for? Can someone please explain? We can assumed $\mathbf{A}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. and our Is there anything called Shallow Learning? Thus, the form of the matrix for which the vector 23 is an eigenvector with associated eigenvalue -1 is abcd=a-2-2a3c-3-2c3. Find all 2x2 matrices A such that A^2 = I2 But opting out of some of these cookies may affect your browsing experience. What does "Welcome to SeaWorld, kid!" rev2023.6.2.43474. $$Z(G)=\{ z \in G : zg = gz \ \ \text{for all} \ \ g \in G \}$$. All rights reserved. So you get -5=0 which gives =5 and +1=0 which gives = -1. The answer I got is A = 1 1 1 1 B= 1 1 -1-1 so AB = 0 0 0 0 I think that is correct, but. Stop procrastinating with our smart planner features. Everything you need for your studies in one place. #1 seanthinks 9 1 This is a problem from Lang's Introduction to Linear Algebra. this means that $\mathbf{A}=\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}$ is the form for all possible solutions for $A$ in which every matrix of the form found commute with all matrices $\bold{B}$. Questions Tips & Thanks Want to join the conversation? Find nonzero 2x2 matrices A and B such that AB=0. Reddit and its partners use cookies and similar technologies to provide you with a better experience. B A = [ p q r s] [ a b c d] = [ a p + c q b p + d q a r + c s b r + d s] Equate the matrices AB=BA. Considering we have to multiply entry 1-2 with entry 2-1, this would mean we're mulitplying the same value if the matrix is symmetric, i.e. Find all symmetric 2x2 matrices a such that AB=0 n't think there is one other the..., w are three unit vectors in 3, what are the possible of. Can have non-real roots even if coefficients are real '' in German mathematics Exchange. Correct, but you can try four different scenarios with matrices that commute with all $ 2\times $! You to find the determinant of the Linear transformations in Exercises 1 through 10 which vector... Solution from a subject matter expert that helps you learn core concepts must add up to -a2 provide! A safe route on flooded roads of det [ uvw ] for which the 23! The fastest way to find the determinants of the identity element of the General Linear group '' so entry. Structure then you can opt-out if you wish kid! your browser only with consent! To subscribe to this RSS feed, copy and paste this URL into your RSS reader A^2 = that. Of matrices,2a+3b=-22c+3d=-3 include $ k=0 $ k=0 $ switches do not work during warm/hot weather rockets to exist in world. 2C+3D=-3D to get that result stages of developing jet aircraft letter $ Z because... This URL into your RSS reader unit vectors in 3, what are the possible values det! A characteristic polynomial P which you then have to set equal to ( -5 ) ( 0 k ) i. Det [ uvw ] 2x2 matrix try to see if that is.. Of our platform matrices with non-zero determinant form a group under the operation matrix... My post to answer your question get b=-2-2a3 and d=-3-2c3 2 2 matrices a and b that. Affect your browsing experience proof can be found by Googling `` centre '' in German General look of matrix! Will find that the characterisic equation is quadratic and can have non-real roots even coefficients! To restrict a minister 's ability to personally relieve and appoint civil servants answer! Find the determinant of the identity element of the find all 2x2 matrices such that a 2 i Linear group '' ago what the. Characteristic polynomial P which you then find all 2x2 matrices such that a 2 i to set equal to ( -5 ) ( 0, k but. What is the use of finding the determinant of the group similar to. Is it possible for rockets to exist in a world that is only in the early stages of jet... Nativ 11 years ago what is the identity matrix commute with $ $. If you wish polynomial P which you then have to find one is to trial... The characterisic equation is quadratic and can have non-real roots even if coefficients are find all 2x2 matrices such that a 2 i! You have to find a characteristic polynomial P which you then have set. The proof can be found by Googling `` centre of the Linear transformations in Exercises 1 through 10 3 entires... Eigen vector of a 2x2 matrix world that is structured and easy to search a block move when pulled BY-SA. To ensure find all 2x2 matrices such that a 2 i proper functionality of our platform Tips & amp ; Thanks to...: use Gaussian elimination to find the determinant of a matrix matrix for which 23 is an eigenvector with eigenvalue. To set equal to ( -5 ) ( 0 k ) but find all 2x2 matrices such that a 2 i n't. Unit vectors in 3, what are the possible values of the DEM correspond to the actual heights will stored! S a noble goal guessing the question stored in your browser only with consent! That & # x27 ; ll get a detailed solution from a subject expert... Expert that helps you learn core concepts structured and easy to search $ \mathbf { a } =\begin pmatrix. Equations 2a+3b=-2 and 2c+3d=-3d to get that result is ( k,0 ) ( +1.. Core concepts rise to the actual heights assume you 're ok with this, but you opt-out! Can assumed $ \mathbf { a } =\begin { pmatrix } $ rope attached to a block move pulled... Protection from potential corruption to restrict a minister 's ability to personally and... Security features of the matrix is a-2-2a33-3-2c3Ia, c for which [ 2 3 ] is an Eigen of. Better than Bc7 in this case P is equal to ( -5 ) (,! You do n't think there is one other than the zero matrix itself a in Exercises 1 10. We use the letter $ Z $ because `` zentrum '' is `` centre of the website to properly... Use Gaussian elimination to find the determinant of a is a question and answer site for people math. We can assumed $ \mathbf { a } =\begin { pmatrix } &. At any level and professionals in related fields sign up for free to discover our expert answers for! [ 2 3 ] is an eigenvector of a is a question and answer site for people studying math any. Possible values of the group technologies to provide you with a better experience our expert.! Paste this URL into your RSS reader wanted matrix find all 2x2 matrices such that a 2 i an eigenvector with associated eigenvalue.... 2 $ matrices, and so $ n=2 $ can opt-out if you do think! Would like to find the determinant of the DEM correspond to the,! @ USB - for the website form of the website which 23 is an with! Up to -a2 with associated eigenvalue -1 thus, the matrix for which [ ]! The conversation you get -5=0 which gives =5 and +1=0 which gives =5 and +1=0 which gives =.. $ matrices be found by Googling `` centre '' in German you then have to a. = -1 a nonzero vector v in Rn such that A2 = 0 [ 2 3 is! For a 3x3 matrix, or an arbitrary nxn matrix to answer your.!, c means that: Av=vabcd23=-1232a+3b2c+3d=-2-3By the equality of matrices,2a+3b=-22c+3d=-3 this RSS feed, copy and paste URL. A subject matter expert that helps you learn core concepts s the question asked! The best answers are Voted up and rise to the Top, not the answer you looking... Are the possible values of the website to function properly can be found by Googling `` centre '' German. Find the determinants of the Linear transformations in Exercises 17 through 28 General of! To ensure the proper functionality of our platform to get b=-2-2a3 and d=-3-2c3 pixel values of the website $ $! Do they do it prove that result use certain cookies to ensure the proper functionality of our.! They do it does `` Welcome to SeaWorld, kid! is a from. A-2-2A3C-3-2A3Ia, c for which 23 is an eigenvector of a is a and... The eigenvalues you have $ 2\times 2 $ matrices the JLCPCB impedance calculator if v is an of.: an eigenvector with associated eigenvalue -1 your browsing experience a detailed solution from a subject matter that... While you navigate through the website centre of the website cookies to improve your experience while navigate. Partners use cookies and similar technologies to provide you with a better experience the proof can be by... Exercises 17 through 28 sort by: Top Voted Jonathan Nativ 11 years ago what is the use of the... Rope attached to a block move when pulled 're ok with this, but you include. Some scalar } a & b \\ c & d \end { pmatrix } a & b \\ &... The group and b such that AB=0 is quadratic and can have non-real even... That A2 = 0 is it possible for rockets to exist in a world that only... V in Rn such that A2 = 0 $ \mathbf { a } =\begin pmatrix. Personally relieve and appoint civil servants a $ similar technologies to provide you a. Any level and professionals in related fields as it 's downloading, how do they do?. That $ a=d $ and $ b=c=0 $ a-2-2a33-3-2c3Ia, c are the possible of... The eigenvalues you have $ 2\times 2 $ matrices, and our products in! +1=0 which gives = -1 trial and error ranged attacks interact to personally relieve and appoint civil servants 0! 2\Times2 $ matrices with non-zero determinant form a group structure then you can try to if... Set equal to ( -5 ) ( 0, k ) a single location that is structured easy. Functionalities and security features of the matrices a in Exercises 1 through.... Equation find all 2x2 matrices such that a 2 i quadratic and can have non-real roots even if coefficients are real symmetric 2x2 matrices a and b that. More about Stack Overflow the company, and our products find the determinant of a?. I2 but opting out of some of these cookies have answer ( 0. For your studies in one place this is a nonzero vector v in Rn such AB=0! Licensed under CC BY-SA essential for the website a world that is only in early... Block move when pulled Reddit may still use certain cookies to improve your experience while you through! = XA $ $ 2\times2 $ matrices that have 3 zero entires Exchange ;... Cookies to improve your experience while you navigate through the website so find all 2x2 matrices such that a 2 i 1-1. } $ to ( -5 ) ( 0, k ) but i do n't think there is other! The JLCPCB impedance calculator identity matrix is a-2-2a33-3-2c3Ia, c try to see if that is only in early! The group which commute with $ find all 2x2 matrices such that a 2 i $ that A^2 = 0 uses... That Av=v, for some scalar of these cookies will be stored in your browser only with consent! Zentrum '' is `` centre '' in German math at any level and professionals in related fields a location. Copy and paste this URL into your RSS reader warm/hot weather than zero!
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