Determinants Definition and 166 Threads

Homework Statement Without expanding the determinant show that bc a^2 a^2 b^2 ca b^2 c^2 c^2 ab = bc ab ca ab ca bc ca bc abHomework Equations 3. Attempt at solution Well, one thing I noticed is that the diagonal row all contain the same values (bc, ca, ab) Using...

Homework Statement This is a practice problem where the solutions are given. Both are 3x3 matrices. det A=-2 and det B=1 find the following: 1)det(A6) 2) det(B-1A3B3AT) 3) det(4(AT)2(B-1)4) 4) det((2BT)-1) Homework Equations The Attempt at a Solution I get the first two...

I'm doing a proof, and near the last step I want to write the expression, \frac{d}{dt} \det{A(t)} = \lim_{\epsilon \to 0} \frac{\det{(A+\epsilon \frac{dA}{dt})} - \det{A}}{\epsilon} which produces the right answer, so I believe that it may be correct. This looks very much like a Taylor...

Homework Statement For which values of x is the matrix (see attachment) invertible? Homework Equations Row ops. Cofactors etc.. The Attempt at a Solution Well, a matrix is only invertible when it's determinant is non zero. I've begun doing some row ops and have just hit a little...

Homework Statement Let A and P be square matrices of the same size with P invertible, Prove detA=det(P-1AP) Homework Equations Suppose that A and B are square matrices of the same size. Then det(AB)=det(A)det(B) The Attempt at a Solution detA=det(P-1AP) detA=det(P-1PA) detA=det(IA)...

I had a question about computing determinants and just was wondering what was allowed. So I know that for an n x n matrix, you can go across a row and choose the matrix element as your determinant coefficient for the (n-1) x (n-1) determinant and you go across the row and do this until you're...

Here is the question: Here is a link to the question: LINEAR ALGEBRA: if A is 3x3 and detA=2 find det(A^-1+4adjA)? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Consider the matrices A = a1 a2 a3 b1 b2 b3 c1 c2 c3 and B = 3a1 4a2+5a1 6a3 3b1 4b2+5b1 6b3 3c1 4c2+5c1 6c3 How are the determinants of A and B related? DO NOT COMPUTE det(A)! Homework Equations The Attempt at a...

Determinant -- best way of introducing determinants on a linear algebra course What is the best way of introducing determinants on a linear algebra course? I want to give real life examples of where the determinant is applied.

Hello I have a question, I think I solved it, and I would like to confirm... Let A be a 4X4 matrix with and let rank(A)=4. It is known that det(A^2) = det(-A) Is det(A)=-1 ? I think the answer is no. det(A)*det(A)=det(-A) det(A)*det(A)=(-1)^4 * det(A) = det(A) det(A)*det(A) = det(A) this is...

Its easy to come with the idea of matrices. Its just a representation of data. But how did the concept of determinants come up? The way we expand determinants with alternate plus and minus sign and then multiplying with the co-factors - how did that come up?

Homework Statement Prove for an operator A that det(e^A) = e^(Tr(A)) Homework Equations The Attempt at a Solution I have no idea how to start. Can someone give me a hint? In general the operator A represented by a square matrix, has a trace Tr(A) = Ʃ A (nn) where A (nn) is...

I understand how to calculate a vector cross product. I also understand that in order to calculate a vector cross product we are calculating the determinant of a 3 x 3 matrix. I guess I have a few questions. 1)How did mathematicians or physicists discover how to calculate the determinant...

Homework Statement P=\begin{pmatrix}3 & -1\\ 2 & 4 \end{pmatrix} Q=\begin{pmatrix}4 & -1\\ -2 & 1 \end{pmatrix} R=\begin{pmatrix}3 & -3\\ 2 & 4 \end{pmatrix} S=\begin{pmatrix}4 & 7\\ 9 & 1 \end{pmatrix} PX = Q QY = R RZ = S Find Matrices X, Y, and Z. Homework...

Let A be an nXn real matrix (a) show that if the transpose of A equals -A, and n is odd, then the determinant of A is 0. (b) show that if (A*A)+I=0, then n must be even. (c) if all the values of A are either 1 or -1, show that the determinant of A is divisible by (2^n-1). these are hard...

I am wondering how one would find a the determinant of a 4x4 or greater. This isn't an urgent question, just a curiosity.

Let M be a transformation matrix. C is the matrix which diagonalizes M. I'm trying to use the formula D = C-1MC. I noticed that depending on how I arrange my vectors in C, I can change the sign of the determinant. If I calculate D using a configuration of C that gives me a negative value for...

I know how to use determinants to solve a system of linear equations, I know I can use them to find the rank of a matrix and find out if a system is linear dependant/independant. However, I still don't really "get" determinants. To me they're some sort of magic box that I can use to calculate...

Homework Statement I'm trying to understand a proof of the LC-KD identity involving determinants (see attachment), from the book Introduction to Tensor Calculus and Continuum Mechanics by Herinbockel. What is the author saying in the last line of text? How can we sum the deltas in the upper...

How important are determinants in a first course on Linear Algebra. In some books it is covered very early after an introduction to matrix algebra but in others it is left to the chpater before eigenvalues and eigenvectors. What is the most appropriate place to situate this topic considering...

In some literature on linear algebra determinants play a critical role and are emphasized in the earlier chapters. (See books by Anton & Rorres, and Lay). However in other literature it is totally ignored until the latter chapters. (See Gilbert Strang). How much importance should we give the...

1. The problem Prove that | (a+b-c) (-c+a-b) (a+b+c) | | (a-c) (c-a) (b-a) | = (a+b-c)(-c+a-b)(a-c) | (a-b) (a-c) (a+b) | using properties of determinants without expanding a determinant 2. The attempt at a solution I tried a lot of...

This is what the symbols in the question represent( sorry about the syntax) ; sr = s subscript r a^r = alpha to the power of r b^r = beta to the power of r g^r = gamma to the power of r Question: If sr = a^r + b^r + c^r, by expressing the determinant as the product of two determinants...

Do we know how we came up with the idea of matrices and determinants? How was the idea of solving linear equations using matrices and determiannts come up. I do not find it useful at all. Does anyone know a site which explains its history and usefulness?

Homework Statement Use the properties of the determinant of a matrix to show that\begin{vmatrix}1+x^2 & x & 1 \\ 1+y^2 & y & 1 \\ 1+z^2 & z & 1\end{vmatrix}=(x-y)(x-z)(y-z) Homework Equations Properties of determinants. There's 10 of them, according to my notes.The Attempt at a Solution I used...

Homework Statement i can't for the life of me find out where the negative comes from! if you have the following the following determinant to calculate det (0) (1) (3) (4) (2) (3) (1) (1) (4) (-1) (-1) (2) (6) (4) (-1) (1)and the next step shows... it equals (note...

Sup guys, So, I'm totally new to mathematica. I need to use it in order to compute a determinant of a 4x4 matrix that is made up entirely of functions. I almost managed to do this in wolfram alpha, but for a 4x4 matrix, the input is too long. Do you guys know how to do this (and if it even...

Hi. I have the following sentence: \begin{array}{l} A,B \in {M_{nxn}}\\ A \ne 0\\ B \ne 0\\ {\rm{if }}AB = 0{\rm{ then}}\\ {\rm{|A| = 0 or |B| = 0}} \end{array} I know this is true but how can I realize? Just thinking about an example? Thanks!

Disclaimer: If this is the wrong place for this, I apologise, this probably comes somewhere between QM, Atomic, Linear algebra and a spoonful of Quantum chemistry for good measure. Anyway, for a group of non interacting (mean field) electrons, moving in a potential generated by nuclei and...

Homework Statement Use Cramer's rule to solve the linear system.Homework Equations (only showing one, I think if one is explained I will figure out the rest) 2x - y = -2 x + 2y = 14 What I'm told I'm supposed to do, is to take the constants accompanying the variables and make a matrix out of...

Homework Statement Prove the chain rule for Jacobi determinants \frac{d(f,g)}{d(u,v)} * \frac{d(u,v)}{d(x,y)}=\frac{d(f,g)}{d(x,y)} Homework Equations Definition of Jacobi determinant \frac{d(f,g)}{d(u,v)} = \frac{d(f,g)}{d(u,v)} = det \begin{bmatrix} \frac{df}{du}&\frac{df}{dv} \\...

The determinant of a matrix is given by the well-known formula det(A) = sump parity(p) * producti = 1...n Ai,p(i) where the p's are all permutations of 1...n and A is a n*n matrix. Parity is +1 for an even permutation and -1 for an odd one. For a permanent, replace parity(p) with 1. For...

I know this may be a very stupid question, but I would really like to know. Is the determinant and the characteristic polynomial of an equation unique? I did several textbook questions and when I look at the solutions, they end up with completely different answers. Sometimes I am wrong and see...

Homework Statement If the entries of A and A^-1 are all integers, how do you know that both determinants are 1 or -1? Homework Equations The Attempt at a Solution I know that 1 = det I = detAA-1=detA * detA-1= detA*(1/detA) = 1 Not sure how we get to - or the role integers...

The lecturer said that a way to find the determinant of a matrix is to do the following det(A) = xdet(B) (1) where A is the original matrix, B is an arbirtray matrix and x is a scalar multiplier The lecturer also said that a simple way to find the determinant of a high...

I'm having a problem with this rule in general. Apparently one can calculate the determinant by multiplying the cofactors and entries of any row or any column of a matrix. I have a negative that pops up. I'll take a 3X3 matrix for simplicity. A= |a b c| |d e f| |g h i|...

Hello guys I've asked to prove following equation on determinants, here it is; Using the properties of determinants & without expanding prove that, see attachment, I need to verify my answer can some one tell me whether is this correct or not?:smile:

How can I make something like determinants tangible? Are there real life examples where determinants are used?

Thanks, although I still haven't managed to factorise the expression although I did type it up in LaTeX! Homework Statement Prove by induction that the following statement is true for all positive integers n. If \lambda is an Eigenvalue of the square matrix A, then \lambda^n is an eigenvalue...

I am to find the general solution of the following two equations, using operator notation: x''-3y'-2x=0 y''+3x'-2y=0 The book suggests starting out with: (D^2 -2)x - 3Dy = 0 3Dx+(D^2 - 2)y = 0 but for the life of me, I do not see how they got this from the first two equations.

I never really enjoyed learning the theory of maths and generally tried to avoid it at all costs since leaving University. However I'm looking at learning kinematic and dynamic chassis modelling and it requires extesive use of vectors and their transformations. I can follow the problems in...

How to prove that ||An||=|A|n2? This property is used in my book but they did not give any explanation/proof of it. Can someone help? Edit: n2=n2

Homework Statement For the determinant \left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right| , b and c being the base of a parallelepiped why is the equation \vec b \cdot (a_1^{'}e_1 + a_2^{'} e_2 + a_3^{'} e_3) = 0 (same goes for vector c)...

Hi, I'm scratching my head over the statement from my textbook which states when determinant is non-zero, the set of vectors blah blah is a basis for r^3. That does not make any sense to me because I know when a row of zeros in a matrix occur; the determinant is zero (through Gaussian...

Homework Statement I was absent and missed the lecture on Cramer's rule and Determinants and have no idea how to start the homework.. The directions and problem are as follows: Using Cramer's Rule, set this problem up to find "a". Only evaluate the Denominator. When finished with the...

Homework Statement I'm supposed to write a proof for the fact that det(A)=det(B) if A and B are similar matrices. Homework Equations Similar matrices have an invertible matrix P which satisfies the following formula: A=PBP^{-1} det(AB) = det(A)det(B) The Attempt at a Solution...

Homework Statement If S is a parallelepiped determined by v1=(1, 1, 0) and v2= (3, 2, 1) and v3=(6, 1, 2) and T: R3--> R3 by T(x)=Ax, find the volume of T(S) Homework Equations {volume of T(S)}=|det A|.{volume of S} The Attempt at a Solution A is [v1 v2 v3] and the |A| = 9 by my...

Hello all, I have been studying some linear algebra, and I recently came upon the method of finding determinants by row reduction (to row echelon form). But isn't it true than a matrix can have any row echelon form? If so, this would mean different determinants, right? I am studying from...

Homework Statement Find the Determinant. \left[\begin{array}{cccc}5&3&0&6\\4&6&4&12\\0&2&-3&4\\0&1&-2&2\end{array}\right]Homework Equations The Attempt at a Solution I'm not sure why I can't get the determinant of this one right. I chose to use the left most column to expand on because the...

Homework Statement This is a problem in differential equations. Find the operational determinant and solve the equation. x'= 4x + y + 2t y' = -2x + y Homework Equations The Attempt at a Solution I'm at a total loss. All the examples in the book have problems with the form: (D - 4)x + 3y =...