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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Using determinants to find a unique solution

I am supposed to find the values of $k$ for which the following system has a unique solution: $$kx+y+z=1\\x+ky+z=1\\x+y+kz=1.$$ I came up with this: ...
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1answer
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Given a Percentage, Find the Smallest Integer Dividend for an Unknown Number of Integer Tests

In a classroom exam where each question is worth 1 point and the number of exam questions are unknown. If the student receives results of a test in a percentage with decimals, let's say ...
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How can I demonstrate that det(C) ≥det(A)?

Let A and B be n×n matrices. and let C=t A +(1-t) B convex sum with 0 ≤t ≤ 1. How can I demonstrate that det(C) ≥det(A) ?
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2answers
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Can we use $R_1 \to R_2-R_1$ as an elementary row operation without changing the value of the determinant?

Can someone help me out? I need to know if $R_1 \to R_2-R_1$ is a valid elementary row operation that can be used on the given determinant without changing the determinant's value. It's a $3\times 3$ ...
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49 views

Can we interchange one row and one column in a determinant?

Can we swap the ith row and the ith coloumn in a determinant as an elementary operation? What happens when we do interchange them? Does the value of the determinant remain constant or does it change?
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1answer
45 views

Find The Determinant Of A Finite Field Matrix

Let there be $A_n=\left(\begin{matrix}4&2&\cdots&2\\2&4&\ddots&\vdots\\\vdots&\ddots&\ddots&2 \\2&\cdots&2&4\end{matrix}\right)\in ...
4
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1answer
55 views

Invertible block matrix

Could I find $A_1,A_2,A_3,A_4 \in M_n(\mathbb C)$, such that, for all $z_1,z_2,z_3,z_4\in \mathbb C$, $\det(z_1A_1+z_2A_2+z_3A_3+z_4A_4)=0$ and $\det \begin{pmatrix} ...
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Proving properties about cofactors when matrix is not invertible

(1) $cof(A^t) = cof(A)^t$ (2) $cof(A)^t = det(A)I$ I have (at least, I think so) proofs of (1) and (2). But the proofs require the matrix $A$ to be non-singular. How do I prove (1) and (2) if $A$ is ...
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Deducing a Recursion formula for Vandermonde Matrix

Vandermonde matrix, $V_n(a_1, \dots, a_n)$ = $\left|\begin{array}{cccc}1 & a_1 & a^2_1 & ... & a^{n-1}_1 \\... & ... & ... & ... \\1 & a_n & a_n^2 & ... & ...
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Confusion about how the determinant changes when all rows are multiplied by a scalar

I am having some trouble thinking about properties of the determinant. I understand why it is true that if $B$ is the matrix obtained from an $n \times n$ matrix $A$ by multiplying a row by a scalar ...
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19 views

How do you solve a determinant thats set to a value with 1 unknown variable?

I was just wondering if someone could explain the steps you take to solve a determinant that has an unknown variable, and is set to equal integer value? For example: How is one supposed to go about ...
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2answers
43 views

Computation of determinant for Using Inverse Function Theorem

Let $f : \Bbb R^{3} \setminus \{(0, 0, 0)\} → \Bbb R^{3} \setminus \{(0, 0, 0)\}$ be given by $f(x, y, z) = (x/(x^{2} + y^{2} + z^{2}), y/(x^{2} + y^{2} + z^{2}), z/(x^{2} + y^{2} + z^{2}))$. Show ...
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1answer
54 views

On the determinant of a certain matrix over the polynomial ring of $n$ variables over a field

Let $A = k[x_1,\dots, x_n]$ be a polynomial ring over a field $k$. Let $\sigma_1,\dots,\sigma_n$ be distinct permutations of the set $\{1,\dots,n\}$. Is the determinant det$(x_{\sigma_i(j)})$ ...
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1answer
15 views

Determinants and row operations

So multiplying a row by a constant multiple the determinant by the same constant and swapping 2 rows will multiply the determinant by a negative right and adding or subtracting does not change the ...
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10 views

finding determinant using elementary matrices

use elementary matrices to find the determinant of the matrix A A = (2 5 -1),(-1 -1 5), (3 7 -3) solution I found: -36
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1answer
24 views

Prove the determinant of A equals 2 times the determinant of B

Question: Prove that $$ det \begin{bmatrix} a + b & p + q & u + v \\ b + c & q + r & v + w \\ c + a & r + p & w + u \\ \end{bmatrix} ...
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1answer
380 views

Matrix with integer entries

The determinant of a given square matrix $A$, with rational entries, equals 1. It is known that all entries of $A^{2015}$ are integers. Is it true that all entries of $A$ are integers? My attemt: ...
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Showing a simple determinant property

I want to use Laplace expansion to show/prove to myself formally that the rule for determinants stating that if B is a matrix obtained from A , where $A \in \mathbb M_{nxn}$, by multiplying a row or ...
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93 views

Find Determinant

How can I find the determinant of a following matrix in the simplest way ? $$\begin{vmatrix} 1 & 10 & 100 & 1000 & 10000 & 100000 \\ 0,1 & 2 & 30 & 400 & 5000 ...
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1answer
29 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
29 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
16 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
24 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 ...
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1answer
38 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
21 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
45 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
67 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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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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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
59 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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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
39 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
26 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
11 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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45 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
43 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
35 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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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
42 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
40 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
47 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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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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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 $B$ ...
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1answer
103 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
137 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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35 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
56 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
38 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 ...