Determinants Definition and 166 Threads

How do we show that, given a matrix $A$, the sign of the determinant is positive or negative depending on the orientation of the rows of A, with respect to the standard orientation of $R^n$?

Ok, so I understand the method of finding a determinant of any order by expansion of minors. I was recently challenged by my teacher to find the determinant of a 10th order determinant she gave me. I succeeded, and felt quite proud of myself, after working for 3 months and filling up 300 pages...

we can solve non-homogeneous equations in matrix form using Cramer's rule. This rule is valid only if we are replacing the columns. Why can't we replace the rows and carry on the same? For eg we can use elementary transformations for obtaining inverses either via rows or via columns. But we...

Homework Statement A is an nxn matrix. Suppose A has the form (^{U}_{W}^{V}_{X}) in which U, V, W, X are n1xn1, n1x n2, n2xn1 and n2xn2 matrices respectively, such that n1 + n2 = n. If V=0, show that detA = detUdetX Homework Equations detA := \sum _{\rho\in sym(n)} sign (\rho)\Pi ai...

I need help with this problem, i am totally lost. See attachment. Please explain.

Homework Statement First Question. How many even Permutations does a 5x5 matrix have? In other words how many permutations are there that would make it +1 instead of -1. Second Question. v= (3,2) w= (4,1) use determinants to find the area of a triangle with sides v, w and v+w...

Homework Statement Suppose A is a square matrix of size n. When is det(-A) = -det(A)? Homework Equations N/A The Attempt at a Solution My approach to the problem is to simply multiply the size n identity matrix by -1, then multiplied by A. For example...

1. The problem statement For integers m >= n, Prove det(xIm - AB) = xm-ndet(xIn - BA) for any x in R. Homework Equations A is an m x n matrix B is an n x m matrix The Attempt at a Solution I tried working out the characteristic polynomials by hand but it just seems too tedious...

Homework Statement Show that two similar matrices A and B share the same determinants, WITHOUT using determinants 2. The attempt at a solution A previous part of this problem not listed was to show they have the same rank, which I was able to do without determinants. The problem is I...

Let A, B and C be 3x3 invertible matrices where det(A)=−4 ,det(B)=−2 and det(C) is some non-zero scalar. Find: det[−2(A^2)^T x C^2 x B^−1 (C^−1)^2] So (A^2)^T is just A^2 since the transpose's det is the same. (C^-1)^2 = C^-2 C^-2 * C^2 = 1 (so just canceled it out) Inverse of B = 1/-2...

Homework Statement Suppose A is a 33 matrix such that det(A)=15. Then det[A3((adj(A))−1)2]= and det[5A−1(adj(A))] -1=inverse Homework Equations I know the properties of determinants and inverses The Attempt at a Solution Problem simplifying to get a number.

I'm having trouble understanding where this concept comes from: Step 1) If you start out with the following two equations v + log u = xy u + log v = x - y. Step 2) And then perform implicit differentiation, taking v and u to be dependent upon both x and y: (d will represent the partial...

New here, have an assignment concerning Cramer's rule which I think I have a decent understanding of - I can use it to find determinants - but am a little lost on a few questions. Given the set of linear equations: a11 x1 + a12 x2 + a13 x3 = 0 a22 x2 + a23 x3 = 0...

Hello, I have several 2x2 matrices, A_s, indexed from s=1 to 50. I need to take the product of their determinants raised to the -1/2 power, i.e., \Pi|A_s|^{-1/2} Can this problem be simplified any further?

Homework Statement I need to find the range for y in the quadratic x^2+2x+3=y Using the determinant b^2-4ac where ax^2+bx+c Homework Equations x^2+2x+3=y The Attempt at a Solution Okay, so: To use the determinant y=0, x^2+2x+3=0, in which case b^2-4.a.c = -8 So there are no real...

Homework Statement detA = 0 Matrix A = | (x+5) 4 4 | | -4 (x-3) -4 | | -4 -4 (x-3)| The Attempt at a Solution I know that if one row or column is equal to another, then detA = 0, so using the last 2 rows, i can find out that...

Homework Statement Express the determinant as a product of four linear factors. \left( \begin{array}{ccc} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \end{array} \right) I'm sure that the only way to do this without hurting yourself is to operate on the determinant and take...

Homework Statement Let A be an nxn matrix, and suppose A has n real eigenvalues lambda_1, ...lambda_n repeated according to multiplicities. Prove that det A = lambda_1...lambda_n Homework Equations None The Attempt at a Solution Could someone explain what is meant by 'repeated...

Homework Statement Prove det [a+p b+r c+s; d e f; g h i] = det [ a b c; d e f; g h i] + det [p r s; d e f; g h i] Homework Equations none The Attempt at a Solution i'm not sure how to prove this though its seems obviously true =S

Homework Statement Let P be an invertible nxn matrix. Prove that det(A) = det(P^-1 AP) Homework Equations none The Attempt at a Solution P^-1 AP gives me a diagonal matrix so to find the determinant , i just multiply the entry in the diagonal. However, i don't understand why P^-1...

Theorem 4.3:The determinant of an nxn matrix is a linear function of each row when the remaining rows are held fixed. That is, for 1<=r<=n, we have det( a1, ..., a_(r-1), u+kv, a_(r+1), ..., a_n ) = det( a1, ..., a_(r-1), u, a_(r+1), ..., a_n )...

Homework Statement If the determinant of a 3 x 3 matrix A is det(A) = 10, and the matrix B is obtained by multiplying the third row by 8, then det(B) = ___? If the determinant of a 5 x 5 matrix A is det(A) = 9, and the matrix C is obtained from A by swapping the second and fourth rows, then...

Homework Statement If A and S are n x n matrices with S invertible, show that det(S-1AS)=det(A). [HINT: Since S-1S=In, how are det(S-1) and det(S) related?] Homework Equations The Attempt at a Solution Not sure. The only thing I can think of doing is substituting S-1S=In into...

I don't think this goes under H/W questions, as it's not a specific question needing solving, or a proof, etc. Getting back to the point, anyone know any good websites or sources that give a good explanation of determinants? I mean what they do, why they do it, not just how to do it. I googled...

a slater determinant gives an asymmetric wave function for fermions is the inverse right? i.e., can the wave function of some fermions always be written in the form a slater determinant? To make things concret, can the sum of two slater determinants be put into the form of a new slater...

Homework Statement let v = (1,0,1) and u = (0,2,1) Find the area of the parallelogram {sv + tu : 0 <= s, t <=1) Homework Equations The Attempt at a Solution I know the area of a parallelogram is the determinant of a 2x2 matrix, but they gave v and u in R^3. Would I just...

If a matrix is diagonalizable, how does its trace equal the sum of its eigenvalues? I can't find a proof for this anywhere.

determinants and the positive and negetive parts off the equation. ive had a couple say they change when they are worked out. on a second order process off mulitiplying diagonally 1 (-1 -11) - -3(1 -11) + -3(2 -1) (1 5) (3 5)...

Homework Statement Prove that [1 a b -a 1 c -b -c 1] is invertible for any real numbers a,b,c Homework Equations A is invertible if and only if det[A] does not equal 0. The Attempt at a Solution I'm not sure if I'm going about this in the correct way; Would I prove this...

Homework Statement If det\left[ \begin {array}{ccc} a&1&d\\ \noalign{\medskip} b&1&e\\ \noalign{\medskip} c&1&f \end {array} \right]=-4 and det\left[ \begin {array}{ccc} a&1&d\\ \noalign{\medskip} b&2&e\\ \noalign{\medskip} c&3&f \end {array} \right]=-1 , then...

Why am I not allowed to reuse rows to find the determinant via elementary operations? Hi, I am learning about matrices and determinants and there is something I can't figure out, straight to the point with an example: Evaluating the determinant... \begin{bmatrix} 1&2&3&4 \\ 5&6&7&8...

I did post earlier about creating a course on linear algebra for myself but got no reply, so i probably will work through the chapter of a book by Riley, Hobson and Bence. However, if you could, please express how you satisfied yourself with such things as matrices and determinants as useful...

Hi, I'm reading a paper where the determinant of the following matrix is solved for using some kind of recurisve method. The matrix is given by M_{ij} = A \delta_{i,j} - B \delta_{i,j-1} - C \delta_{i,j+1}, with i,j = 1...N and are NOT cyclic. The author sets D_N =...

Hi, I'm trying to prove the theorem found in the following scan of Coleman's Aspects of Symmetry. So far, I have managed to show that the functions on both sides have simple zeros/poles, but I fail to see why they are meromorphic functions, and the steps that lead to their asymptotic...

I'm trying to learn about Determinants and Cramer's Rule. If a multiple of one row is added to another row, the value of the determinant is not changed. This applies to columns, also. 15 14 16 18 17 32 21 20 42 Factoring a 3 from C1 and a 2 from C3 = 6 times 5 14 13 6 17...

I'm really putting some effort into understanding the core idea behind determinants. For a 2x2 matrix, I obviously saw how to derive the formula for the determinant (using AA^-1=I). The question is, how did they define the |A| for higher order matrices? I'm reading a textbook on linear algebra...

I think I have something mixed up so if someone can please point out my error. 1. the set of all linear combinations is called a span. 2. If a family of vectors is linearly independent none of them can be written as a linear combination of finitely many other vectors in the collection. 3. If...

I know I am not presenting an actual problem, but it is for homework, and I do need some help. I wasn't sure which forum to post in, so I posted in two. :( Sorry. I am doing a presentation on the 3-point problem in Geology. We have to use Cramers Rule to solve for the equation of a plane. I...

Homework Statement Hi, could someone please confirm my results. I just put my answers because the procedure is so long. let me know if you get the same results. 1) Wronskian(e^x, e^-x, sinh(x)) = 0 2) Wronskian(cos(ln(x)), sin(ln(x)) = 1/x * [cos^2(ln(x)) + sin^2(ln(x))] = 1/x thanks in...

Homework Statement A 2 x 2 matrix B satisfies B (3 1)^{T} = B (5 2)^{T} What is det (B) ? Give a reason Homework Equations None really The Attempt at a Solution I really have no idea how to start solving this. Does it involve inversing?

[b]1. The proof that det(kA) = k^ndetA where A is nxn I read somewhere that det(rI(n)) = r^(n) so det(rA) = det(rI(n).A) = r^ndetA but I am really confused about how they got that? Is I the identity matrix? What would the det(I) be?

I'm having trouble relating the cross product form |a||b|sin(theta) to its component form (a1b2 - a2b1) ... and so on... I know how to do this mathematically so please don't just suggest some proof that I can find in every textbook... The component form involves the solutions to equations...

For an n x n matrix A, what is the relationship between det(A) and det(-A)? I tried it with a 1x1 matrix, and det (-A) = - det (A) I tried it with a 2x2 matrix, and det(A) = det(-A) I tried it with a 3x3 matrix, and the results were the same as that with a 1x1. This leads me to believe...

Homework Statement I hope that I'm going to make sense here. I found the Jacobian of two functions by taking the partial of F1 w.r.t x, then y and same for F2. My professor did it by taking the partial of F1 w.r.t. y, then x, same for F2. So I have a 2x2 matrix. When i go to find the...

Homework Statement Consider a dynamical system x(t+1) = Ax(t),, where A is a real n x n matrix. (a) If |det(A)| > or equal to one, what can you say about the stability of the zero state? (b) If |det(A)| < 1, what can you say about the stability of the zero state? Homework Equations...

Everyone is going to start to hate me know because i keep asking so many questions, but really i do look around these boards to help someone, but the questions are to hard for me or someone answers before me! :redface: anyway my question is: If the determinant of a b c d e f g h...

Is the fact that \vec{a}\times\vec{b} = \det \begin{bmatrix}\hat{e}_1& \hat{e}_2 & \hat{e}_3 \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{bmatrix} just a coincedence (ie. a mneminic device) or does it have some deep mathematical significance? edit: Also, is the cross product, like the dot...

hi all, have a rather sticky problem on determinants to deal with...hope someone can offer some help. [In what follows, the numbers in brackets denote suffixes, so that, for example, A(s)(j) refers to the element in the sth row of A and jth column of A.] Let A be an n x n matrix. Let s be a...

A=[3 2 4 3 ;2 -1 2 -2 ; 1 2 0 -2 ;-2 -5 -5 -4] can smby pls show me how to perform this determinants pls thanx

hi, i seem to have some trouble proving: Suppose M = [A B:O C], where A is a kxk matrix, C is a pxp matrix, and O is a zero matrix. Show that det(M) = det(A)det(C). my attempt at a proof: det(M) = det(A)det(C) det[A B:O C] = det(A)det(C) AC - OB = det(A)det(C) AC =...