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
23 views

Does cofactor expansion generalize to complex matrices?

When finding the determinant of some $n * n$ matrix $A$ when $$\forall i,j\in\mathbb{N} ,i\leq n\land j\leq n\implies A_{ij} \in \mathbb{C}$$ Can cofactor expansion be used under the normal definition ...
0
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2answers
35 views

Show determinant equals 0

Ok, i've been working on the following problem and this is what I've gotten: Let $F$ be a field, let $n$ be a positive integer, and let $A,B \in M{nxn} (F)$ be matrices satisfying $B\ne 0$ and ...
0
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0answers
38 views

Computing characteristic polynomial of tridiagonal block matrices

I want to compute the characteristic polynomial of symmetric matrices of the form \begin{bmatrix} A & U & & & 0\\ U & B & V & &\\ & V& C& W &\\ ...
0
votes
3answers
88 views

for which a, the matrix A is diagonalizable?

A = $ \begin{pmatrix} 2a+3 & 0 & 0 \\ -a-3 & a & a+3 \\ a & a & a+3 \\ \end{pmatrix} $ Characteristic polynomial: $ ...
15
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9answers
919 views

Why do determinants have their particular form?

I know that for a matrix $A$, if $\det(A)=0$ then the matrix does not have an inverse, and hence the associated system of equations does not have a unique solution. However, why do the determinant ...
22
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2answers
1k views

Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby ...
0
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2answers
57 views

How to prove determinant is a group homomorphism and onto?.

I posted this question I am struggling with previously but it was put on hold for lack of context, I hope this is now clearer. Consider the determinant function Det: Mn($\mathcal{F}$) $\to$ ...
2
votes
2answers
95 views

Calculate $\lvert A \rvert$ if $a_{ij}=0$ if $i=j$ and $1$ otherwise [duplicate]

Let $n$ be a positive integer and let $A=[a_{ij}] \in M_{n\times n} (R)$ be the matrix defined by $a_{ij}=0$ if $i=j$ $1$ otherwise To be honest, I've only calculated determinants of matrices ...
0
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2answers
58 views

How find this matrix determinant value

Find the value $$ \det\left| \begin{array}{c&c&c&c&c} 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & AB^2 & AC^2 & AP^2 \\ 1 & AB^2 & 0 ...
3
votes
2answers
106 views

Determinant involving recurrence

Evaluate $$\left| A \right| = \left| {\matrix{ {x + y} & {xy} & 0 & \cdots & \cdots & 0 \cr 1 & {x + y} & {xy} & \cdots & \cdots & 0 \cr 0 ...
13
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4answers
528 views

Expected Value of a Determinant

Suppose that I construct an $n \times n$ matrix $A$ such that each entry of $A$ is a random integer in the range $[1, \, n]$. I'd like to calculate the expected value of $\det(A)$. My conjecture is ...
4
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2answers
109 views

How prove this $|A||M|=A_{11}A_{nn}-A_{1n}A_{n1}$ [duplicate]

Question: let the matrix $A=(a_{ij})_{n\times n},i=1,2,\cdots,n,j=1,2,\cdots,n$, and the matrix $M=(a_{ij})_{(n-2)\times (n-2)},$ mean that $$A=\begin{bmatrix} a_{11}&\cdots&a_{1n}\\ ...
1
vote
1answer
65 views

Compute the determinant $4\times 4$

Compute the determinant: $$ A= \begin{vmatrix} 1 & 1 & a+1 & b+1 \\ 1 & 0 & a & b \\ 2 & b & a & b \\ 2 & a & a ...
1
vote
1answer
48 views

Four coplanar points in $\mathbb{N}^3$ space

Is it possible to write out natural number coordinates of four three-dimensional points $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d} \in \mathbb{N}^3$, with the following determinant zero? ...
0
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1answer
14 views

Evaluating Determinants using elementary operations

I'm having problem regarding the Evaluation of a determinant. Can anyone explain me if there are any rules regarding row operations? I mean which row should I evaluate first?
0
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1answer
24 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 ...
2
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0answers
52 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 ...
4
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2answers
78 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 ...
0
votes
1answer
17 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 ...
1
vote
1answer
24 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 $A$.
0
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2answers
89 views

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

Let A be real square matrix. If $\det (A^2 - I) < 0$, then A has eigenvalue $\lambda \in (-1,1)$. How to prove this?
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0answers
39 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) = ...
0
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2answers
77 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
61 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
votes
1answer
44 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 ...
3
votes
3answers
49 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
votes
3answers
509 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
0
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2answers
55 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
21 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 ...
2
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1answer
55 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 $f$? ...
0
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1answer
47 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 ) ...
0
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5answers
66 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 ...
1
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4answers
122 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} ...
0
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1answer
66 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\\ ...
3
votes
1answer
50 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 ...
8
votes
3answers
90 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 ...
2
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0answers
42 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
24 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 ...
1
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2answers
62 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 $U$?
0
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0answers
46 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 ...
10
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2answers
139 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 + ...
1
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1answer
40 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 $q \in \mathbb{R}[x]$. ...
2
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2answers
62 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 ...
2
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1answer
42 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 ...
1
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1answer
24 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
76 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
53 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
64 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 & ...
1
vote
1answer
70 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 ...
5
votes
1answer
92 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, ...