Question about determinants, computation or theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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35 views

Argument that Vandermonde matrix's determinant has $n-1$ distinct roots

det(Vandermonde) = $\left|\begin{array}{ccccc}1 & x & x^2 & ... & x^{n-1} \\1 & a_2 & a^{2}_{2} & ... & a^{n-1}_{2} \\1 & ... & ... & ... & ... \\1 ...
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1answer
33 views

Finding the eigenvalues of a matrix problem

So I do know how to compute the eigenvalues of a matrix. At least, that's what I thought. I got the matrix A = \begin{bmatrix}1&-2&0\\-2&0&2\\0&2&-1\end{bmatrix} My approach ...
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1answer
31 views

Determinant of an almost-diagonal matrix

I would like to compute the determinant of the $(k+1)\times (k+1)$ matrix below $$J=\begin{vmatrix} y_{k+1}& 0 & \ldots & 0 & y_1 \\ 0& y_{k+1}& \ldots& 0& y_2 \\ ...
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1answer
29 views

Why is the determinant of a basis of a dual space non zero?

Let $P[t]{_2}$ = $V$ a vector space. A basis $B$ = $(1,t,t^2)$ of V and $B$* = ($e_1$,$e_2$,$e_3$) the dual basis of $B$. $f_a$: $V$ $->$ $R$ , $p(t)$ $->$ $p(a)$ (evaluation). Show that ...
5
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1answer
43 views

Proving that $\det(A)R^n \subset\mathrm{Im}(\Phi)$

Let $R$ be a commutative ring with unity and consider the free $R$-module $R^n$. Given a matrix $A$ with coefficients in $R$, define a homomorphism $\Phi: R^n\to R^n$ by $\Phi(u) = Au$. My question is ...
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1answer
23 views

Prove $cof(A^t) = cof(A)^t$

I'm trying to use $A^{-1} = cof(A)^tD$, where $D = det(A)^{-1}$ to prove $cof(A^t) = cof(A)^t$. I end up with statements these two $A^{-1} = Dcof(A)^t$. And $(A^{t})^{-1} = D cof(A^t)$. But I don't ...
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1answer
53 views

Plücker Relation: misunderstanding?

I'm trying to understand exterior algebra better by gaining some "bare hands" understanding of the exterior powers $\Lambda^k(X)$ in more detail when $\dim(X)$ is small. I think so far I understand ...
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3answers
73 views

Proof Regarding Determinants of a Matrix

Prove the following statement: If $A$ is an $n$ by $n$ matrix, such that $\sum_{j = 1}^n a_{ij} = 0$, for all $1 ≤ i ≤ n$, then $\det A = 0$ too. (Sorry I don't know how to format this equation) ...
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0answers
60 views

A determinant identity

I'm looking for a proof of the following identity $$\delta_{\mu_1\mu_2\mu_3\mu_4}^{\nu_1\nu_2\nu_3\nu_4}(AB)^{\mu_1}_{\nu_1}(AC)^{\mu_2}_{\nu_2}(AD)^{\mu_3}_{\nu_3}(AE)^{\mu_4}_{\nu_4} = \det A ...
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2answers
28 views

Question regarding matrices and determinants.

The question states : If $A$ is a non-singular square matrix satisfying $AB-BA=A$, then prove that $|B+I|=|B-I|$. Note : $1.$ Here $I$ is the Identity matrix. $2.$ Modulus sign means determinant. ...
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2answers
21 views

Replacing 3x3 matrix with a value to work out the determinant

Goodday, I need some assistance with the following problem Let |a b c| |p q r| |x y z| = 6 and find ...
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1answer
62 views

Question on linear algebra - Determinant multiplication.

Does anybody have a "non brute" force way to prove the following for non-singular matrices A, B: det(AB) = det(A) det(B)
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0answers
53 views

How can I determine the sign of a term in a determinant when the indices are out of order?

I am reading Shilov's book linear algebra. He explains how to compute determinants. Basically, for the plus terms you write \begin{equation} x_{a1}x_{b2}x_{c3}x_{d4}x_{e5} x_{f6} \end{equation} and ...
4
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1answer
42 views

Matrix of Ones with Diagonal of Integers

My teacher posed a question to the class today asking us to find the determinant of the following matrix... \begin{bmatrix} 2 & 1 & 1 & 1 & 1 \\ 1 & 3 & 1 & 1 ...
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1answer
27 views

How to compute the last diagonal element of a matrix using determinants?

I actually want to verify the following statement. Please note that I am not even sure if it is correct. I tried out some numerical examples in R and it seems that the statement is correct and can ...
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1answer
26 views

Log-Determinant Concavity Proof

Can you please help me understand how he gets the equation marked by red from the above one ?
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3answers
53 views

determinant of the symmetric matrix $8\times8$

How to compute the determinant of the following matrix: $ \left( \begin{array}{cccccccc} 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 & 1 & 1 & ...
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1answer
62 views

Looking for a proof that the resultant is the product of the differences of roots

I'm trying to find a general proof to an exercise given in Garrity et al's book, Algebraic Geometry: A problem-solving approach. The problem is this: Given two polynomials f and g, show that for ...
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1answer
41 views

A determinant coming out from the computation of a volume form

I am convinced that the following identity is true: \begin{equation} \det\begin{bmatrix} 1+a_1^2 & a_1 a_2 & a_1 a_3 & \ldots & a_1a_n \\ a_1a_2 & 1+a_2^2 & a_2a_3 & ...
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0answers
27 views

Determinants, Pfaffians, and…?

I recently stumbled across the wikipedia entry on Pfaffians and found them rather interesting, especially the property below. (assuming $A$ is a $2n\times 2n$ skew symmetric matrix) ...
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1answer
57 views

Eigenvalue of the sum of a symmetric matrix and the outer product of it's eigenvector

I have a symmetric matrix $A$ with eigenpairs $(\lambda_k, v_k)$ with $k \in (1,..,n)$. A new matrix $B$ is made from an eigenpair $(\lambda_i, v_i)$ like this: $$B = A - \lambda_i v_i v_i^T$$ where ...
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1answer
51 views

Change in determinant when multiplying row of a matrix

I'm a bit confused with something I read and I hope you can help me. I'm studying determinants and right now how matrix row operations change the determinants. I read (and in fact quote): the effect ...
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1answer
50 views

Solving a system of equations

(Image Attached) I've begun with the hint and found out that $(det\ A)*x=adj\ A*c$ and therefore what x is. My question would be how would I go about finding what $det\ A_i$ is? Should I go about ...
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22 views

Linear program of 0-1 knapsack problem and proof of integer

I have some questions about the knapsack problem. How can the 0-1 knapsack problem described as a linear program? How to proof that the solution of the 0-1 knapsack problem are integer? (I'm ...
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0answers
79 views

calculation of the determinant of a block matrix little help

I need to prove $$\operatorname{det}\begin{pmatrix}A & B \\ C & D\\ \end{pmatrix}= \operatorname{det}(DA-CB),$$ where $A,B,C,D \in M_{n\times n}(R)$ with the property that $A$ and $B$ ...
5
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1answer
120 views

Upper Triangular Block Matrix Determinant by induction

We want to prove that: $$\det\begin{pmatrix}A & C \\ 0 & B\\ \end{pmatrix}= \det(A)\operatorname{det}(B),$$ where $A \in M_{m\times m}(R)$, $C \in M_{m\times n}(R)$,$B \in M_{n\times n}(R)$ ...
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1answer
166 views

Extending a Chebyshev-polynomial determinant identity

The following $n\times n$ determinant identity appears as eq. 19 on Mathworld's entry for the Chebyshev polynomials of the second kind: $$U_n(x)=\det{A_n(x)}\equiv \begin{vmatrix}2 x& 1 & 0 ...
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1answer
40 views

Find the computational cost associated with calculating determinant of an $n\times n$?

How to determine the computational cost associated with calculating determinant of an $n\times n$ matrix, using LU factorization.
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1answer
61 views

Find the matrix given the determinant

Is there a general method to find a 3x3, or 2x2 matrices, given the determinant? I want to do a project with my students when we start to study Systems of Equations. It would be interesting if the ...
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1answer
30 views

Determinant using Leibniz formula

$$\begin{matrix} * & * & *&*&* \\ *&*&*&*&*\\ 0&0&0&*&* \\ 0&0&0&*&* \\ ...
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1answer
53 views

Cholesky, Inverse, and Determinant when updating the diagonal of a symmetric positive definite matrix

Suppose that $A$ is a symmetric positive definite matrix and assume its dimension $n$ is large. Let $I$ be the $n \times n$ identity matrix and $m \neq 0$ be a scalar. I'm interested in computing as ...
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1answer
42 views

$d+1$ distinct points of a rational normal curve in $\mathbb{P}^{d}$ are linearly independent

Let $X\subset\mathbb{P}^{d}$ be a rational normal curve. After a change of coordinates, it is the image of the map: $\nu:\mathbb{P}^{1}\rightarrow\mathbb{P}^{d}, (a_{0}:a_{1})\mapsto ...
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2answers
44 views

Finding a Matrix from Determinants

I've stumbled upon this problem on my homework, and I have no clue how to do it, and haven't found any help online: If I'm understanding this correctly, then $det(M) = ad - cb + eh - gf$ ? What I ...
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2answers
60 views

matrices vector spaces

Consider the vector space of 3 by 3 matrices with real coefficients. Let W denote the subset of matrices with determinant 0. Decide whether W is a subspace or not.
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2answers
58 views

Why adding a row with another row in square matrix A doesn't change the $\det(A)$ value?

Why adding a row with another multiplied row in square matrix $A$ doesn't change the $\det(A)$ value?
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1answer
57 views

Need help with a determinant problem [closed]

I'm learning determinants and just came across a problem. I've been trying really hard to solve it but no success so far. I just know that the answer is (3) 1 but don't know how to solve it? Please ...
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0answers
34 views

Matrices over $\Bbb{R}$ with $2\times 2$ skew-symmetric blocks

For any complex number $z\in{\Bbb C}$, define a $2\times 2$ matrix $\hat z$ as $$ \hat z:=\begin{pmatrix} a&-b\\ b&a \end{pmatrix} $$ where $z=a+ib$, $a,b\in{\Bbb R}$. Let $A=(z_{ij})$ be an ...
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1answer
38 views

Do we have $\det=e^1\wedge\cdots\wedge e^n$?

If we think of the determinant as a multilinear map from the set of $n$-column vectors to $\mathbb{R}$, $$\det:\mathbb{R}^n\times\cdots\times\mathbb{R}^n\to\mathbb{R},$$ am I right in saying that ...
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27 views

Can the Berezinian be used instead of the determinant of a block matrix?

Suppose we have a $2N \times 2N$ matrix $(N\ge 2)$. For example we can consider a block matrix: $$X= \left[\begin{matrix} A & B \\ C & D \end{matrix}\right]$$ with $A,B,C,D=$ $2\times 2$ ...
4
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1answer
83 views

Series Expansion of the determinant for a matrix near the identity.

The problem is to find the second order term in the series expansion of the expression $\mathrm{det}( I + \epsilon A)$ as a power series in $\epsilon$ for a diagonalizable matrix $A$. Formally we ...
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1answer
17 views

Is the function that maps a matrix to the determinant of a submatrix continuous?

Let $M$ be the space of $m \times n$ matrices over $\mathbb{R}$. For each $A$ in $M$ let $A'$ be a fixed submatrix of $A$. Is the function $M \to \mathbb{R}$ defined by $A \mapsto \det(A')$ ...
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14 views

Recursive equations with matrices, and a question about determinants in relation to power of matrices

If we have the matrix equation $AX^{(i)} = X^{(i+1)}$ where $A$ is a constant matrix, this is what we'd call a recursive function; in matrix form. Moreover, if $X^{(i+1)} = X^{(i)}$, i.e. $AX = X$ ...
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1answer
164 views

Show that a 2x2 matrix A is symmetric positive definite if and only if A is symmetric, trace(A) > 0 and det(A) > 0

I need to show two parts of the implication are true. First: if $A$ is $2\times 2$ and is symmetric positive definite then $trace(A)>0$ and $det(A)>0$. Second: if $trace(A)>0$ and ...
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1answer
29 views

Show that any orthogonal matrix has determinant 1 or -1 [duplicate]

Hello fellow users of this forum: Show that for any orthogonal matrix Q, either det(Q)=1 or -1. Thanks
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1answer
39 views

Determinant of a matrix with 2x2 blocks

I have a matrix, say $A$ and want to find it's determinant $detA$. A is $L\times L$ and made up of $2\times 2$ blocks $M_{i,j}$ giving it a total size of $2L \times 2L$. The entries of the blocks ...
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0answers
14 views

prove that $X$ is invertible if and only if $Y$ is invertible. if $(-1)^i(1+i)x_i^T=y_i$

$X=[x_1,x_2,...,x_n]$ and $Y =$ $y_1\\y_2\\...\\...\\...\\y_n$ where $x_i$ and $y_i$ are column and row matrices respectively. $X$ and $Y$ are both $n$ x $n$ matrices. if $$(-1)^i(1+i)x_i^T=y_i$$ ...
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13 views

Identity with discriminant

Let $K$ be a field which is finite or of characteristic zero, let $L$ be an extension of finite degree $n$ of $K$, and let $σ_1,…σ_n$ be the $n$ distinct $K$-isomorphisms of $L$ into an algebraically ...
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0answers
61 views

Can one define wedge products using determinants for $n$-forms?

I was talking to Ted Shifrin in math chat yesterday and he mentioned there is a way to define wedge products using determinants. As far as I understand, given a set of vectors $x,y,z,v,u... \in ...
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0answers
13 views

General Discriminant Formula for Multivariate Polynomials over Reals.

Consider \begin{eqnarray} b'(I_{s}c-A)b>0 \end{eqnarray} where $A$ is a symmetric, positive definite s by s square and I is the identity and c is a constant. Solutions are to be found using $b ...
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196 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix. I have already asked a (viewed but unanswered) ...