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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10
votes
2answers
138 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
vote
1answer
36 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
votes
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
votes
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
vote
1answer
17 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 &... ...
1
vote
2answers
37 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
votes
1answer
43 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 $F$ ...
1
vote
1answer
51 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
69 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 ...
4
votes
1answer
91 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, ...
1
vote
1answer
32 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 \, ...
0
votes
2answers
25 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 ...
0
votes
1answer
17 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 ...
4
votes
1answer
81 views

Characterization of positive definite matrix with principal minors

A 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, for ...
2
votes
2answers
205 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 ...
0
votes
4answers
43 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
votes
2answers
194 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
votes
1answer
21 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 ...
1
vote
1answer
35 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 $\det M$.
2
votes
1answer
56 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
votes
2answers
103 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
votes
1answer
43 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 ...
0
votes
2answers
76 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: $$ ...
12
votes
1answer
157 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 ...
1
vote
4answers
57 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 ...
2
votes
3answers
63 views

Is there any way to check wheter the determinant of a matrix $A$ with $|\text{det }A|=1$ is positive or negative?

Let $A\in\text{GL}(n,\mathbb{R})$ with $|\text{det }A|=1$. Is there any way to check wheter $\text{det }A$ is positive or negative without computing it?
0
votes
0answers
30 views

Amount of sub-matrices created by Laplace expansion

I have created a program that solves a matrices determinant using the Laplace expansion method, and I was wondering if there is a equation which provides how many sub-matrices are created and used in ...
0
votes
0answers
21 views

Representative value of non-square matrix

First of all, I apologise if this question is inappropriate, I wish I could be more specific - but due to the nature of it, as I am actually asking for a suggestion of some technique, that's hard to ...
12
votes
3answers
195 views

determinant inequality $\det(A^2+AB+B^2)\geq\det(AB-BA)$

$A,B$ are two $2\times 2$ real matrices, then $$\det(A^2+AB+B^2)\geq\det(AB-BA)$$ The inequality is equivalent to the following problem: Let $X=A+\dfrac{B}{2},Y=-\dfrac{B}{2}$ ...
0
votes
2answers
47 views

Odd-dimensional skew-symmetric matrix is singular, even in a field of characteristic 2

I'm familiar with the usual proof $\det(A) = \det(A^T) = \det(-A) = (-1)^n \det(A)$ which only works in fields of characteristic not equal to 2. To get a proof that works in characteristic 2 I can ...
0
votes
1answer
21 views

Determinant of a matrix over a field K

Let $A$ be an $n \times n$ matrix over a field $K$. Do the properties of the determinant of a real matrix hold for the matrix $A$? If not, in which fields do the properties of the determinant of a ...
0
votes
2answers
32 views

finding determinant for matrix using upper triangle method!

so Here an example for matrix that I'm trying to evaluate its determinant! | 1 3 2 1| | 0 1 4 -4| | 2 5 -2 9| | 3 7 0 1| when applying first row operation i get | 1 3 2 1| | 0 1 ...
1
vote
1answer
21 views

meanings determinants of matrices in finite field

Let's $\Bbb{Z}_q$ is finite field. ($q$ is prime number). Lets $A_1$ – set of matrices $n\times n$, such that $\det(M) = 1$, for any matrix $M \in A_1, A_2$ – set of matrices $n\times n$, such that ...
3
votes
2answers
53 views

The number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$

How to count the number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$? I know that the number of invertible matrices is GL$(n,p)$. Have any ideas?
3
votes
0answers
24 views

Determinant of triangular matrix except for one column (atomic/Gauss/Frobenius)

Is there some "smart" way to calculate determinants that look like this? $\begin{vmatrix}-1&a_{1,2}&a_{1,3}&a_{1,4}&\cdots&a_{1,m-1}&a_{1,m} ...
1
vote
2answers
45 views

Find $2\det ( \frac{1}{2} A )$ given that $A$ is $3\times 3$ and $\det(A)= -2$

Here is a question that should be done today: If $A$ is $3\times 3$ and $\det(A)= -2$, find $2\det(\frac{1}{2}A)$. I solved this problem but I am not sure because the way I used is not accurate! ...
7
votes
4answers
259 views

Determinant of a Special Symmetric Matrix

If $A$ is a symmatric matrix of odd order with integer entries and the diagonal entries $0$ then $A$ has determinant value even. I can prove the result if I can show that the eigenvalues of $A$ are ...
2
votes
4answers
104 views

How to prove the inequality $\det (AA^T) \ge 0$?

How to proof for any matrix $A \in \Bbb R^{n \mathbf x n}$, that the next inequality $\det(AA^T) \ge 0$ is true?
3
votes
3answers
66 views

Determining the values of $\lambda$ for which the matrix is invertible

I'm working on a homework problem and am a little stuck. The question is: Determine the values of $\lambda$ for which the matrix $$\begin{pmatrix} \lambda &-1&0\\ -1&\lambda&-1\\ ...
1
vote
2answers
42 views

Explicit formula for inverse of upper triangular matrix inverse

I have $n \times n$ upper triangular matrix $A$ such as $$ \begin{bmatrix} x_1 & x_2 & \ldots & x_n \\ 0 & x_1 & \ldots & x_{n-1} \\ \vdots & \vdots & ...
6
votes
5answers
400 views

Find the determinant of the following;

Find the determinant of the following matrix, and for which value of $x$ is it invertible; Now I don't really know how to procees as I get find a suitable row operations that will simplify ...
0
votes
1answer
24 views

volume of parallelotopes

I know that determinant indicates the volume of a parallelotopes spanned by the n vectors. I absolutely understand that the properties of a determinant: any function $f:\mathbb{R}^{n\times ...
0
votes
0answers
33 views

How to prove this identity? (About determinant)

How to prove the determinant of \begin{equation} \left( \begin{array}{cccccc} a_{00} & a_{01}& a_{02} & \cdots & \cdots& a_{0k}\\ 1 & a_{11}& a_{12} & \cdots ...
0
votes
1answer
26 views

Find $det(xy^T)$ where $x$, $y$ are vectors from $R^n$, $n$>1 [duplicate]

I represented $x$ as $[x_1\ x_2\ ... x_n]^T$, and $y^T$ as $[y_1\ y_2\ ... y_n]$. Multiplying them produces a matrix $n$x$n$: $$ \begin{pmatrix}x_1y_1&x_1y_2&\dots& x_1y_n\\ ...
3
votes
1answer
44 views

Prove positive definiteness

I want to prove that the matrix $$\begin{pmatrix} 1 &\cfrac{1}{2} &\cfrac{1}{3} &\cdots &\cfrac{1}{n} \\ \cfrac{1}{2} &\cfrac{1}{3} &\cfrac{1}{4} &\cdots ...
0
votes
0answers
46 views

Help proving by induction that the determinant is equal to: $(-1)^{\frac{n(n-1)}{2}}\cdot a_{1n}\cdot a_{2n-1} \cdots a_{2n1}$

Help proving that the determinant is equal to $(-1)^{\dfrac{n(n-1)}{2}}\cdot a_{1n}\cdot a_{2n-1} \cdots a_{2n1}$ $$ \begin{vmatrix} 0 &0 & \dots &0 &a_{1n}\\ 0 &0 & \dots ...
2
votes
1answer
17 views

Computing determinants using derivatives in an arbitrary field

When computing determinants that depend on a parameter $t\in \Bbb R$, it is often useful to use the fact that \begin{align} \det(V_1(t),\dots,V_n(t))&=\det(V_1(a),\dots,V_n(a))+\\ ...
2
votes
2answers
50 views

real matrix with $Tr((A-I)^{T}(A-I) )<1$

$A$ is a $n\times n$ real matrix, $$\operatorname{Tr}((A-I)^{T}(A-I) )<1$$ then $\det(A)\ne0$. well, $$\sum_{i\ne j}a_{ij}^2+\sum (1-a_{ii})^2\lt1$$ How to derived $\det(A)\ne0$? Thank ...
1
vote
0answers
47 views

probability of having a non-zero determinant

$K=\mathbb{Z}_p$ for some prime p, and $dim V = n$. It has been shown that the number of different bases in $V$ is: $\frac{1}{n!} \prod_{i=0}^{n-1}(p^n - p^i)$ (bases which are permutations of one ...
-1
votes
1answer
50 views

How to prove that $\det\left[\pmatrix{u_1 & v_1\\ u_2 & v_2\\ u_3 & v_3}\pmatrix{s_1 & s_2 & s_3\\ t_1 & t_2 & t_3}\right]=0$?

Evaluate $\det\left[\begin{pmatrix} u_1 & v_1\\ u_2 & v_2\\ u_3 & v_3 \end{pmatrix} \begin{pmatrix} s_1 & s_2 & s_3\\ t_1 & t_2 & t_3 \end{pmatrix}\right]$. I ...