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But that's not a symmetric matrix. Here's the solution worked out if you want it:https://en.wikipedia.org/wiki/Idempotent_matrix#Real_2_.C3.97_2_case, New comments cannot be posted and votes cannot be cast. Second: what about the determinant? For a given matrix A, we find all matrices B such that A and B commute, that is, AB=BA. That's the question. This chapter is all about transformations. More from my site. My first thought was to multiply the following matrix by itself: Since I need to get back A, I figure I should then set each element of A2 to the original element I need, and I get this: All I can get from these four equations, however, is something I already know from equations 2 and 3: Any other manipulations I try to do just leads me to the same exact equation. Use two different nonzero columns for B. I know I can put some variables in B and then multiply AB and then that equation = 0, but I still can't seem to crack it. Which means you're done with that case - you have two degrees of freedom, a and either b or c. If you backtrack, and say a+d is not 1, then you get a=d (see "Third" above), b=0, and c=0. Eigenvalues and characteristic polynomials? 2×2 determinants can be used to find the area of a parallelogram and to determine invertibility of a 2×2 matrix. So let's suppose b = 0. Let's transform this homogeneous system into a matrix. That's the question. You should analyze those cases as well. It might be that there are no such matrices, of course, but then you'll come up with a contradiction and you'll know there was a mistake. By that method, if you let Let B = . ad-bc = 0 or 1. One way is by doing what you've done and just playing around until you've found something. If A^2 = A, it follows that A^2 - A = 0, where 0 is the null or zaro matrix. Then q q * = q * q = (ad − bc) I, where I is the 2 × 2 identity matrix. Well my first inclination was to find them all so I could become experienced solving these sorts of questions, but I'm not faring too well. A square root of a 2×2 matrix M is another 2×2 matrix R such that M = R 2, where R 2 stands for the matrix product of R with itself. We then get, We then get that a and d are both plus or minus 1. The site may not work properly if you don't, If you do not update your browser, we suggest you visit, Press J to jump to the feed. Solution: Many solutions, but one example is A= 1 1 1 1 1. A good way to double check your work if you’re multiplying matrices by hand is to confirm your answers with a matrix calculator. For a given 2 by 2 matrix, we find all the square root matrices. This is obviously not a valid deduction to make; a = d = 1, b = c = 0 is a solution, for example, so it is not always true that a+d=1. For the case where det(A) = 1, you can easily show that there is only one instance of this. In addition to multiplying a matrix by a scalar, we can multiply two matrices. 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. Ok, so, here's some random looks at the equations, without trying to change your basic approach or anything. To find a 2×2 determinant we use a simple formula that uses the entries of the 2×2 matrix. If a2=1, then you know a = 1 (and not -1) from a (ad-bc) = (ad-bc) , above. Note: If the entries are allowed to be complex numbers, then there are more solutions. squaring it. Thank you. Show that if a matrix is skew-symmetric, then its diagonal entries must all be 0. Note that in this context A−1 does not mean 1 A. 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. Always double-check in the end anyway. Find all symmetric 2x2 matrices A such that A^2 = 0. So there are a few ways you could go about trying to solve this. Theinverseofa2× 2 matrix The inverseof a 2× 2 matrix A, is another 2× 2 matrix denoted by A−1 with the property that AA−1 = A−1A = I where I is the 2× 2 identity matrix 1 0 0 1!. If you're asking to characterize all of them, it depends on how much theory you have, and with more theory there's better explanations. The site may not work properly if you don't, If you do not update your browser, we suggest you visit, Press J to jump to the feed. You get a2 - d2 = a - d, e.g. For instance, if we assume b is 0 we can use the equations to find one example of an idempotent matrix with 0 in the top right. As you can see, (1) = (3), so this system is dependent, and will have infinitely many solutions. a) Show that W is a linear space. So if entry 1-1 is a, the first multiplication is a*a = a2. Ohhhhh. If b_1 and b_2 denote the 2x1 columns of a 2x2 matrix B, then the columns of the matrix product AB are precisely the 2x1 vectors Ab_1 and Ab_2. Let A denote the given matrix. Is the question to find one, or classify all? The 1-2 and 2-1 entries have to be the same in a 2x2 symmetric matrix. Just ways to keep pushing stuff forward. The examples above illustrated how to multiply 2×2 matrices by hand. So if entry 1-1 is a, the first multiplication is a*a = a 2. Let us take the other option: a+d = 0 i.e., a = -d. Now, a^2 = -bc implies c & b are of opposite sign and |c| = a^2/|b|. Factorizing A (A - I) = 0, where I is the identity matrix. Starting with a^2 + b^2 = 0, b(a+c) = 0, and b^2 + c^2 = 0: The quiz is designed to test your understanding of the basic properties of these topics. I have a solution, but I got there through a mistake. The next value to be added to this must add up to -a2. By definition. Post all of your math-learning resources here. So am I right in saying that no symmetric matrix A exists (other than the zero matrix, of course) such that A2 = 0? I don't think there is one other than the zero matrix itself. b) Find all possible values for the dimension of W. 13. Now, some more clever ideas: take equation 1 times d minus equation 2 times c. This makes the resulting right side equal to the determinant: Which turns into either the determinant is 0 (which we already had as a possibility) or a = 1, which is exciting. We want to determine the all values of it so that the matrix nonsingular. squaring it. For a given matrix A, we find all matrices B such that A and B commute, that is, AB=BA. To solve this problem, we use Gauss-Jordan elimination to solve a system Since det(A)2 =det(A), that should give you det(A) = 0 or 1, e.g. It might be that there are no such matrices, of course, but then you'll come up with a contradiction and you'll know there was a mistake. How about image space? New comments cannot be posted and votes cannot be cast, More posts from the cheatatmathhomework community, Continue browsing in r/cheatatmathhomework, Looks like you're using new Reddit on an old browser. All symmetric 2x2 matrices will be of the form. I don't think there is one other than the zero matrix itself. Find nonzero matrices A;B;Csuch that AC= BCand A6= B. The only way I can get it to work is if I use imaginary numbers. So I have this homework problem that I thought would be rather simple and straightforward, but I was wrong. So it's not a valid deduction because I can't divide by a variable unless I'm confident it's non-zero, right? If you assume a+d = 1, and plug that into (EDIT(clarity): your equation 4), you get bc + (1-a)2 = 1-a, which simplifies to bc + a(a-1) = 0; since a-1=-d, this is actually bc-ad=0 for that case, which means that if a+d = 1, the determinant must be 0, e.g. Hence, the zero matrix is the only such matrix. Press question mark to learn the rest of the keyboard shortcuts. so from the first paragraph we get ad = bc so if one of a or d is zero one of b or c is also zero. Since you want to find a nonzero matrix B with the property that AB is the zero matrix, you want to find two column vectors b_1 and b_2, at least one of which is nonzero, satisfying A b_1 = 0 and A b_2 = 0. Third: Subtract eqaution 4 from equation 1. While there are many matrix calculators online, the simplest one to use that I have come across is this one by Math is Fun. so cd = c and d2 = d, obviously d isn't 0 so d is 1 and c is free. B= 1 0 1 0 C= 0 1 0 0 4. For instance, if we assume b is 0 we can use the equations to find one example of an idempotent matrix with 0 in the top right. a could have been - 1 and d 1, or we could have set a different variable equal to 0, or we could have set it equal to something else and seen what happened. Questions, no matter how basic, will be answered (to the best ability of the online subscribers). The next value to be added to this must add up to -a 2. An obvious solution to the third equation, then, is. But no such real value exists whose square will be negative. Have you gotten to vector spaces and basis? Since there are only 2 idempotent square matrices, you can just try them both for parts a and b. The flaw is that ab + bd = b when either a + d = 1 or b = 0. You're very clever! Hi, If you look at your definition of idempotent A^2=A, then you can actually solve this for A and find *all* idempotent square matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. Find all possible values for the dimension of W. 12. Squaring this matrix and setting it to 0, we end up with three equations: a2 +b2 =0 ab+bc=0 b2 +c2 =0 Which values of a,b,c satisfy these equations? The first thing I'll point out is that if you only have to find one such example, the identity matrix clearly works. These obviously aren't the only solutions in this vein either. In linear algebra, a nilpotent matrix is a square matrix N such that = for some positive integer.The smallest such is called the index of , sometimes the degree of .. More generally, a nilpotent transformation is a linear transformation of a vector space such that = for some positive integer (and thus, = for all ≥). You can probably use an induction proof for this case for integers. Let U and V both be two-dimensional subspaces of R5, and deﬁne the set W := U+V as the set of all vectors w = u+v where u ∈ U and v ∈ V can be any vectors. By doing this, I was able to find a matrix that satisfied A2 = A. I'll check with my professor to make sure it's an acceptable approach tomorrow. Is this the right conclusion to make or am I still missing it? Thank you so much for your input. There are many possible solutions. We have not reached those points, no. What about for a 3x3 matrix, or an arbitrary nxn matrix? Press question mark to learn the rest of the keyboard shortcuts, https://en.wikipedia.org/wiki/Idempotent_matrix#Real_2_.C3.97_2_case. If you left multiply by A-1 you get A = I, so now you know A = I or det(A) = 0. now also we get that this equation is not preserved by scalar multiplication, in fact if something is a member of this set it immediately implies all the scalar multiples of that thing are not in this set. Help please. Similarly, if you add equations 2 and 3 together, you get (a+d)(b+c)=(b+c), which, once again, gives you an alternative b+c=0 case (EDIT: This is redundant with the b=c=0 cases, though, so kinda not useful). Always double-check in the end anyway. Assuming you don't need to find every such matrix, just one, a trick you can do is try setting some of these variables equal to 0.