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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1answer
55 views

Differentiation involving determinant

This question has arisen by following the proof in the appendix of Louis Liporace's paper on maximum-likelihood estimation, where the paper concerns classes of probabilistic functions (elliptically ...
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0answers
60 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
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2answers
95 views

Determinant and trace as conjugations?

For real matrices $A$ it holds that $$\det\,\big(e^A\big)=e^{\mathrm{tr}\,A}$$ so we can write $$\mathrm{tr}=(\exp)^{-1}\circ \;\det\;\circ\;(\exp).$$ Is this interpretation of trace as the ...
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1answer
29 views

Taking product of cofactor with different row

Given a matrix $A=(a_{ij})_{n\times n}$, let $C_{i,j}$ be the cofactor in position $(i,j)$. By the determinant formula, we have $$\det A=\sum_{i=1}^n a_{i,1}C_{i,1}.$$ What about if we take a ...
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51 views

Determinant of matrix of linear transformation in complex vector space

Let $V$ be finite complex vector space, $a\not= 0$ an element of $V$, and $f$ linear functional on space $V$. $A: V \to V$ has definition: $A(x)= x - f(x)*a$. Find determinant of $A$.
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127 views

Proving that $\det (A^2 - I) < 0 \Rightarrow \lambda \in (-1,1)$

Let $A$ be real square matrix. If $\det (A^2 - I) < 0$, then $A$ has an eigenvalue $\lambda \in (-1,1)$. How to prove this?
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79 views

Complex matrices: looking for homomorphism

Let $\mathbb{C}$ denote the complex numbers, and let $M_2(\mathbb{R})$ be the ring of $2$ by $2$ matrices with real entries. Define a function $f:\mathbb{C} \to M_2(\mathbb{R})$ by $ f(a+bi) = ...
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2answers
146 views

Prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$

I want to prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$ If $A\in \mathbb{F}^{m\times n}$ and $B\in \mathbb{F}^{n\times m}$ It is easy to show that $0$ has algebraic multiplicity of at least $m-n$ ...
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2answers
73 views

How can det(B)=-det(A) when this happens?

There's a property that says when you interchange two rows/columns from a matrix A, the resulting determinant B will have its determinant equal to the original one, but with its sign inversed: ...
4
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1answer
48 views

Maximal determinant of a matrix filled with $\pm 1$

Is there an algorithm to determine what is the maximal determinant you can get just by putting $1$ or $-1$ in a square matrix? For example in a $3\times3$ matrix: $$ \begin{bmatrix}1 && -1 ...
4
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3answers
54 views

$M$ matrix, $\mathrm{rank}\ M=1$. Prove that $det(e^M)=1$ iff $M$ is not diagonalizable

M is a $n\times n$ matrix over $\mathbb R$. with $\mathrm{rank}\ M=1$. Prove that $det(e^M)=1$ if and only if $M$ is not diagonalizable. I really don't know how to start thinking about this.. :/ I'd ...
5
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3answers
1k views

Is $\det(AB) =\det(BA)$

I am having trouble proving if $$ \det(AB) = \det(BA) $$ is right or wrong. $A,B$ are square matrices. Can you please point me to the right direction? Thank you
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2answers
160 views

Cubic roots of determinant.

If x=a+2b satisfies the cubic (a,b element of R) f(x)= $$\left|\begin{matrix} a-x & b & b \\ b & a-x & b \\ b & b & a-x\end{matrix}\right|$$ =0, then it's other 2 roots are?
0
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1answer
25 views

Divisibility of determinant.

If the three digit numbers: $x17, 3y6, 12z$ where $x,y,z$ are integers from $0-9$ are divisible by a fixed constant $k,$ then the determinant $$\left|\begin{matrix} x & 3 & 1 \\ 7 & 6 ...
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1answer
65 views

What can be said about functions of constant Hessian determinant?

Let $f:\mathbb{R}^2\to \mathbb{R}$ with $\det \nabla^2f = 1.$ Let's also assume that $\nabla^2 f$ is positive-definite (which we can do WLOG by adjusting the sign of $f$). What can we say about $f$? ...
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1answer
133 views

Angle between 2 vectors using the determinant

I have a polygon like this: I basically want to find the angles $\alpha$, inside the polygon, between the vectors. I'm using the determinant to calculate the angle alpha: $det(\vec V2, \vec V2 ) ...
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5answers
69 views

If $A =\begin{pmatrix} -1 & 0 & 1\\ 0 & 1 & 1\end{pmatrix}$ and $AB = I$ find the $3\times 2$ matrix $B$.

Alright so you multiply $A$ and $B$ and you get four equations. Then you do $\det[AB] = \det[I] = 1$ and you get a fifth. I'm stuck here now. What else can I do to find $B$? I'm trying to get this ...
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4answers
147 views

How to prove the following exercise by using the definition of a determinant?

$\begin{align} \begin{vmatrix} a_{11} & \cdots& a_{1m} & 0 & \cdots & 0 \\ \cdot & \cdots & \cdot & \cdot & \cdots & \cdot \\ a_{m1} & \cdots & a_{mm} ...
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1answer
111 views

How to find the determinant of a NxN matrix

Here is my matrix. How do I find the determinant of this one? I'm really trying to solve it but I can't think of anything. $$ \begin{pmatrix} 3 & 2& ...& 2\\ 2& 3& ...& 2\\ ...
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1answer
145 views

Determinant (or positive definiteness) of a Hankel matrix

I need to prove that the Hankel matrix given by $a_{ij}=\frac{1}{i+j}$ is positive definite. It turns out that it is a special case of the Cauchy matrices, and the determinant is given by the Cauchy ...
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3answers
126 views

Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue.

I'm taking a linear algebra course and the professor included the problem that prove $$ \rm{det}(I+\epsilon A) = 1 + \epsilon\,\rm{tr}\,A + o(\epsilon) $$ Since the professor hasn't covered the ...
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0answers
55 views

Differential Equations and Eigenvalues

I have the following system of differential equations: $$\left\{\begin{aligned} \frac {dx} {dt}=-4x+2y \\ \frac {dy} {dt}=-\frac 5 2x+2y \end{aligned} \right. $$ Which corresponds to the following ...
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1answer
88 views

Bound on the degree of a determinant of a polynomial matrix

I want to implement a modular algorithm for computing the determinant of a square Matrix with multivariate polynomials in $\mathbb{Z}$ as components (symbolically). My idea is first to reduce the ...
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2answers
64 views

Determinant of $U$, Determinant of $U^T$

Given an $n\times n$ matrix $U$ such that $U^TU = I_n$, the $n\times n$ identity matrix. Then what are the possible values of the determinant of $U$?
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54 views

The determinant of a matrix

In order get the determinant of$$\begin{pmatrix} \lambda-n-1 & 1 & 2 & 2 & 1 & 1 & 1& 1 & \cdots &1 & 1 \\ 1 & \lambda-2n+4 & 1 & 2 & 2 &2 ...
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2answers
181 views

Prove or disprove : $\det(A^k + B^k) \geq 0$

This question came from here. As the OP hasn't edited his question and I really want the answer, I'm adding my thoughts. Let $A, B$ be two real $n\times n$ matrices that commute and $\det(A + ...
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1answer
88 views

determinant of an endomorphism

Let $n \in \mathbb{N}$. We define: $f_n:\mathbb{R}[x]_{\leq n} \rightarrow \mathbb{R}[x]_{\leq n}, p\mapsto(p \cdot x)' $, where $q'$ is the derivation of a math. polynomial $q \in \mathbb{R}[x]$. ...
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2answers
68 views

How to calculate this determinant? [closed]

$F(x) = x(x - 1)(x - 2) \cdots (x - n + 1)$ good morning I want to prove that I want to calculate the determinant $$\left| \begin{array}{*{20}c} {F(a)} & {F'(a)} & {F''(a)} & \cdots ...
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1answer
54 views

Find the differential of $f(A)=det(A^{-1}-A)$ where $A$ is invertible.

The question is if $A$ is an invertible matrix with real entries of size $n$. Is $f(A)=det(A^{-1}-A)$ differentiable? and what is the differential. I think I managed to show it's differentiable. the ...
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1answer
49 views

determinant of special structure block matrix

How do you compute the determinant of the block matrix: $$ M = \begin{bmatrix} A+B & A &A &A &... \\ A & A+B &A &A &... \\ A & A &A+B &A &... ...
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2answers
892 views

Determine the values of $k$ so that the following linear system has unique, infinite and no solutions.

Determine the values of $k$ so that the following linear system has a unique solution, infinite solutions and no solution. $2x + (k + 1)y + 2z = 3$ $2x + 3y + kz = 3$ $3x + 3y − 3z = 3$ I have ...
0
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1answer
208 views

Jacobian matrix and determinant - relation to orientation

$F$ is a function from $V$ to $V$ where $V$ is a $n$-dimensional vetor space and $p \in V$. In the article Jacobian determinant it says: "If the Jacobian determinant at $p$ is positive, then $F$ ...
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1answer
124 views

Trace of the exterior power as a determinant

Let $A$ be a matrix. According to Wikipedia, $$tr(\wedge^k A) = \frac{1}{k!} \det \begin{pmatrix} tr (A) & k-1 & 0 & \cdots \\ tr (A^2) & tr (A) & k-2 & \cdots \\ \cdots & ...
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1answer
74 views

Does there exist $B$ for which $BB^T=I$?

My question is Does there exist a real matrix $B_{n\times m}$ with $m<n$ for which $BB^T=I_n$? Why do I need this? Suppose we are given a real matrix $Q_{m\times n}$ (again, with ...
6
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1answer
119 views

Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} - \det M_{ij}\det M_{ji}$

Let $M$ be a (real) $n \times n$ matrix. For $1 \leq i, j \leq n$ we denote by $M_{ij}$ the $(n-1) \times (n-1)$ matrix that we get when the $i$th row and $j$th column of $M$ are removed. Now, ...
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1answer
570 views

Determinant with Levi-Civita Symbol?

From Schaum's Outline in Tensor Calculus If $A = [a_{ij}]_{nn} $ is any square matrix, then define $\text{ det } A = \epsilon_{i_1i_2i_3...i_{n-1}i_n}a_{1 \, \cdot \, i_1}a_{2 \, \cdot \, ...
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2answers
30 views

Evaluating determinant of an implicit matrix

I know that row operations does not change the determinant of a matrix but I also know that for example, A is a nxn matrix and if det(A) = 2 then, det(2A) = (2^n)*det(A). So, how should I approach ...
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1answer
32 views

Matrices in systems of linear equations

I've been working on matrices lately. Currently, I am stuck on solving systems of linear equations using matrices. I've read the following article which has proved very helpful in understanding the ...
5
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1answer
668 views

Characterization of positive definite matrix with principal minors

A symmetric matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, ...
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2answers
207 views

Why this representation of circle is valid?

A line passing through two distinct points $P_1(x_1,y_1),P_2(x_2,y_2)$ can be expressed by $$\det\left| \begin{array}{ccc} x-x_1&y-y_1 \\ x_2-x_1&y_2-y_1 \\ \end{array} \right|=0$$ Since ...
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4answers
67 views

Set of all matrices with determinant 0, non-zero

I was assigned this problem in class: Let $f: M(n, \mathbb R) \rightarrow \mathbb R $ be given by $f(X) = det(X)$. Identify the sets $f^{-1}(0)$ and $f^{-1}(\mathbb R^*)$, where $\mathbb R^*$ denotes ...
6
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219 views

Determining the derivation of a determinant

Let $\Phi\colon E\to M$ with $E\subset \mathbb{R}\times M$ and $M\subset\mathbb{R}^n$ open. Consider the function given by $x\mapsto \Phi(t,x)$ for fixed $t\in\mathbb{R}$. (1) Determine $$ ...
0
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1answer
33 views

Proof regarding effect of row operations on determinants>

Let $A,B \in K^{n,n}$ and suppose $B$ is obtained from $A$ by adding $\lambda$ times row $j$ to row $i$. Prove $det(A)=det(B)$. My Attempt I tried to use proof by induction for this . Take ...
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1answer
45 views

Find determinant $\det M$, where $m_{ij}=a_ia_j$, and $m_{ii}=a^2_i+k$

Let ${a_1,\dots,a_n}$ --- sequence and $k\ne 0$. Define matrix $M$ in following way: $m_{ij}=a_ia_j$ if $i\ne j$, and $m_{ii}=a^2_i+k$. Find $\det M$.
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1answer
85 views

Coordinate-free proof of determinant of transpose

I'm interested in a coordinate-free proof of the statement $\mathrm{det}(A) = \mathrm{det}(A^T).$ Let $V$ be a finite-dimensional vector space over a field $K$, and let $f : V \rightarrow V$ be an ...
7
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2answers
127 views

How to find determinant of this matrix?

Is there a manual method to find $\det\left(XY^{-1}\right)$ ? Let $$X=\left[ {\begin{array}{cc} 1 & 2 & 2^2 & \cdots & 2^{2012} \\ 1 & 3 & 3^2 & \cdots & 3^{2012} \\ ...
2
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1answer
79 views

If first 1 by 1 upper left submatrix (principal minor) = 0, conclude straightaway saddle point ? - Question 8

Find all local extremal points for the function $f(x,y) = x^3 - 3xy+y^3 $ and classify their type. For $H(f)(0,0),$ I see that $D_1 = \det [0] = 0$. So according to the criteria that I already posted ...
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2answers
336 views

Inverse of a sum of positive definite matrices

Let $A,B$ be symmetric positive definite matrices. Let $A^{-1} = LL^T$ (Cholesky decomposition, $L$ is lower-triangular). I think the following identities are true, but I haven't found them online: $$ ...
13
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1answer
224 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
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4answers
68 views

Find matrix determinant

How do I reduce this matrix to row echelon form and hence find the determinant, or is there a way that I am unaware of that finds the determinant of this matrix without having to reduce it row echelon ...