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
855 views

To find trace and determinant of matrix [duplicate]

Possible Duplicate: Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$ Let $A$ be a $4\times 4$ matrix with real entries such that $-1,1,2,-2$ ...
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2answers
158 views

Is the determinant of a matrix lower when all its elements are lower?

Problem Consider a generic matrix $A$, we are going to think of a simple case by taking into consideration a $3 \times 3$ matrix: $$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} ...
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2answers
161 views

Determinant of matrix composition

Considering a matrix given by composition of square matrices like: $$ M = \begin{pmatrix} A & B\\ C & D \end{pmatrix} $$ I want to calculate its determinant $|M|$. Consider that all ...
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2answers
974 views

Slick proof the determinant is an irreducible polynomial

A polynomial $p$ over a field $k$ is called irreducible if $p=fg$ for polynomials $f,g$ implies $f$ or $g$ are constant. One can consider the determinant of an $n\times n$ matrix to be a polynomial in ...
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2answers
75 views

Determinant of symmetrical factorized matrix

Given $A, B \in \mathbb{R}^{n\times n}, t \in \mathbb{R}\setminus \{0\}$ with $b_{ij} = t^{i-j}\cdot a_{ij}$. Prove $\det(A) = \det(B)$. I first thought of induction. I can easily prove this for $n ...
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2answers
229 views

Interesting Determinant

Let $x_1,x_2,\ldots,x_n$ be $n$ real numbers that satisfy $x_1<x_2<\cdots<x_n$. Define \begin{equation*} A=% \begin{bmatrix} 0 & x_{2}-x_{1} & \cdots & x_{n-1}-x_{1} & ...
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4answers
578 views

Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$

Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues. If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be ...
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1answer
172 views

Determinantal criterion for positive semi-definite matrix

Consider an $n\times n$ real matrix $K$ which satisfies $$ \det[K_{ij}]_{i,j=1}^k\geq 0,\qquad 1\leq k \leq n. $$ I know that if one assumes moreover that $K$ is symmetric, then $K$ is positive ...
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1answer
195 views

What is $\frac{\det(A+tI)}{\det(B+tI)}$ as $t\to0$?

If $A$ and $B$ are two real $2\times 2$ matrices with $\det A = 0 $ and $\det B = 0 $ and $\mathrm{tr}(B)$ is non zero. then what will be limit of $$\lim_{t\to0}\frac{\det(A+tI)}{\det(B+tI)}$$ I used ...
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1answer
157 views

Can we say that there exist an integer n such $A+nB$ invertible?

If $A$ and $B$ are $3\times 3$ matrices and $A$ is invertible, then can we say that there exist an integer $n$ such that $A+nB$ invertible? I was trying by choosing n such that eigne values of $A+nB$ ...
4
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1answer
180 views

$A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to 1. find the possible value of the determinant

Let $A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to $1$. And remaining nine entries of the row being $0$. Which of the following is not a possible value of the ...
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4answers
158 views

Determinant of a matrix $A$ is zero when its has a zero submatrix of dimentions $p \times q$ and …

Let $A$ be a $n \times n$ matrix and suppose $A$ has a zero submatrix of order $p \times q$ where $p + q \ge n+1$. Then $\det(A) = 0$. I can see this happening when doing Laplace expansion. I can ...
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1answer
2k views

determinant of a sum

I need a formula for the determinant of the sum of two matrices: $\det(\mathbb{I}+M)$. On the internet I found it for the first order but i need it at second or even third order. Where can I find the ...
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1answer
3k views

Determinant of the sum of matrices

Let D be a diagonal matrix and A a Hermitian one. Is there a nontrivial way to calculate the determinant of A from the determinant of A+D and the entries of D? It can be assumed that the diagonal ...
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2answers
269 views

Determinant form of equation, 3 variables, third order (nomogram)

I'm trying to put the following equation in determinant form: $12h^3 - 6ah^2 + ha^2 - V = 0$, where $h, a, V$ are variables (this is a volume for a pyramid frustum with $1:3$ slope, $h$ is the height ...
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1answer
2k views

Are complex determinants for matrices possible and if so, how can they be interpreted?

I've been asked to compute the determinant of a 3x3 matrix with complex entries. I have done so using the normal expansion along a row or column method that I would use were the entries real. My ...
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2answers
288 views

Linear Algebra - Finding Eigenvalues of a Matrix

$A=\begin{bmatrix}3 & -2 & 5\\ 1 & 0 & 7\\ 0 & 0 & 2\end{bmatrix}$, Find the eigenvalues of A. I realized that if I swap columns I and II then I can make it an upper ...
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1answer
66 views

Area of a geometric configuration

How to find the area of the triangle in the plane R2 bounded by the lines y=x, y=-3x+8 and 3y+5x=0. How can I solve this? I'm thinking i can take y=x as the origin and just use y=-3x+8 and 3y+5x=0 ...
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1answer
187 views

Proving that an $n\times n$ matrix has at most $n$ distinct eigenvalues

$A$ is a $n\times n$ matrix over the field $F$. How can I prove that there are at most $n$ distinct scalars $c$ in $F$ such that $\det(cI - A) = 0$? Thank you!
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1answer
138 views

Is there a name referring to this result?

For any real $m \times n$ matrix $A$, it seems that $$\det(I_n + A^{T}A) = \det(I_m + AA^{T}) $$ always holds, where $I_n$ is the identity matrix of size $n$. Though I have not tried to prove this ...
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3answers
90 views

Determinants of Matrices and Their Properties

I tried gaussian elimination and ended up with: $\begin{bmatrix} v1\\ v2\\ v3\\ \frac{5}{2}v1+v4 \end{bmatrix}$ Then I used the rule that says ...
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2answers
437 views

Elementary proof that if $A$ is a matrix map from $\mathbb{Z}^m$ to $\mathbb Z^n$, then the map is surjective iff the gcd of maximal minors is $1$

I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by an $m \times n$ matrix $A$, then the map is surjective iff the gcd ...
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1answer
60 views

In a matrix-/vector-equation $\small A*x=B*y$ must $\small \operatorname{sign}(\det(A))=\operatorname{sign}(\det(B))$?

I've a matrix-equation and I'm trying to list the conditions under which that equation can be true. The equation is of the form: $$\small \begin{pmatrix} -A & B+b \\ A+a & -B \end{pmatrix} ...
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1answer
413 views

Proof of $\det(AB)=\det(A) \det(B)$: confused about $(c\alpha_{i}+\alpha_{i})B$

I am currently studying for a final exam and am confused about the proof of $\det(AB)=\det(A)\det(B)$ given in Hoffman/Kunze. I'll type out the entire thing so that my question will be in the correct ...
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1answer
344 views

Meaning of this 4x4 determinant

Let $p,q,r$ and $s$ be four points on the plane. Moreover, $p,q,r$ are given in clockwise order. My book said that the following determinant is positive if and only if $s$ lies inside the circle ...
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1answer
337 views

Elementary row operations effect on determinants

I have a matrix $$ A = \begin{pmatrix} 1 & 2 & 3 & 4\\ -1 & 1 & 2 & 3\\ 1 & -1 & 1 & 2\\ -1 & 1 & -1 & 1\\ \end{pmatrix} $$ I should be ...
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2answers
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Use of determinants

I have been teaching myself maths (primarily calculus) throughout this and last year, and was stumped with the use of determinants. In the math textbooks I have, they simply show how to compute a ...
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105 views

Problem on the determinant of a Matrix

Suppose we have $n-1$ linearly independent vectors $a_1, \ldots,a_{n-1} \in \mathbb{Z^n}$. Is it possible to find another vector $a_n\in \mathbb{Z^n}$ such that the determinant of the matrix $M$ ...
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3answers
174 views

Different approaches to evaluate this determinant

How to evaluate this determinant $$\det\begin{bmatrix} a& b&b &\cdots&b\\ c &d &0&\cdots&0\\c&0&d&\ddots&\vdots\\\vdots ...
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1answer
99 views

Definition of the 1-dimensional $\mathbb{C}GL(V)$ module “$\det ^n$”

I'm reading through my notes on representation theory of $S_n$ and $GL(V)$, and have come unstuck on a definition which I can't understand - furthermore I can't seem to find any information on it ...
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3answers
352 views

Matrix - Show $\det(A) =0$

I am a little stuck on this Matrix problem. Suppose that for complex square matrix A,B the following holds: $AB -BA = A$ Show that $\det(A)=0$ That would mean that A has no inverse. So I thought, ...
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1answer
144 views

Number of matrices with weakly increasing rows and columns

I'm curious as to how many matrices there are of size $m \times n$ with elements of the set $\{1, \ldots , k\}$ such that each row and column is weakly increasing? The answer should be expressable as ...
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0answers
118 views

Determinant expression for the power sum

Let $S_{n,r} := \sum_{k=1}^{n} k^r$ be the power sum. On the homepage by W. Hecht (link) I have found the following determinant expression: $$S_{n,r} = (-1)^{r-1} \frac{n(n+1)}{(r+1)!} \det ...
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2answers
236 views

Charts, spheres and determinants

Here are some things I don't understand, I would be very grateful for any help! I am trying to find a chart on the unit sphere that preserves area. The most natural map that springs to my mind is ...
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1answer
993 views

Some theorem about block matrix determinants with symmetric inner matrices?

I could do this problem with bruteforce but I think there must be some elegant theorem that helps to calculate the determinant with the block matrix (here having symmetric matrices inside) such as: ...
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2answers
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Determinant of Large Matrix with Gauss rule?

$$A=\begin{pmatrix} 1 & -1 & 0 & 2 \\ 2 & 1 & 0 & 0 \\ 1 & 1 & 2 & 2 \\ 0 & 0 & 1 & 1 \\ \end{pmatrix}$$ With the lower determinant method, I ...
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1answer
72 views

Determinant of square matrix multiplied on right and left by rectangular matrices

Let $A$ be an $n\times n$ matrix with, say, real entries. Let $B$ be $m\times n$, where $m<n$. If the determinant of $A$ is known, can we say anything about $\det(BAB')$? What if we put ...
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0answers
113 views

Determinant of the Laplacian of a surface is this correct?

given a surface with metric $ g_{ab} $ i would like to evaluate the functional determinant of the Laplacian in the form $ - \partial _{s} \zeta (0,E^{2})=\log\det( \Delta + E^{2}) $ then i need to ...
7
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2answers
115 views

Determinant of matrix $(x_j^{n-i}- x_j^{2n-i})_{i,j=1}^{n}$

Good evening all, I am determined to determine this determinant: $$D = \det{\left[x_j^{n-i} - x_j^{2n-i}\right]_{i,j=1}^{n}}$$ Looking at the smaller cases, leads me to believe that $$D = \prod_{1 ...
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1answer
141 views

question in linear algebra, matrices

Given $A$ and $B$, $2\times 2$ matrices, which of the following is necessarily true? If $A$ and $B$ are both Unitary matrices over $R$ and $\det(A)=\det(B)=1$ then $A$ is similar to $B$. If $A$ and ...
2
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2answers
167 views

finding the determinant

Is there any fast way to compute the determinant of this matrix $$ \left( \begin{array}{ccccc} a & b & 0 &0 &0 \\ b & a & b &0 &0 \\ 0 & b & a &b ...
5
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3answers
257 views

computing determinant of a matrix

let $A$ be an $n\times n$ matrix with entries $a_{ij}$ such that $a_{ij}=2$ if $i=j$. $a_{ij}=1$ if $|i-j|=2$ and $a_{ij}=0$ otherwise. compute the determinant of $A$. using the famous formula ...
4
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1answer
877 views

Quick ways to _verify_ determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors …

What are easy and quick ways to verify determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after calculating them? So if I calculated determinant, minimal ...
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2answers
399 views

Showing $\prod\limits_{i<j} \frac{x_i-x_j}{i-j}$ is an integer

Let $x_1,...,x_n$ be distinct integers. Prove that $$\prod_{i<j} \frac{x_i-x_j}{i-j}\in \mathbb Z$$ I know there is a solution using determinant of a matrix, but I can't remember it now. Any ...
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2answers
276 views

Finding inverse of a $3\times 4$ or $4\times 3$ matrix

Now I have no problem getting an inverse of a square matrix where you just calculate the matrix of minors, then apply matrix of co-factors and then transpose that and what you get you multiply by the ...
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1answer
432 views

How to prove a Wronskian identity?

The following Wronskian identity can be proved by expanding both sides and checking that two sides are the same. But how to prove it more elegantly? Let $u_1(x), u_2(x), u_3(x), u_4(x)$ be four ...
3
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76 views

Identifying factors of higher order in a determinant

Consider a $n\times n$ matrix $A$ whose elements are some polynomials in the indeterminates $x_1, x_2,\ldots,x_m$. To calculate the determinant of such a matrix, one of the usual ways is to treat the ...
3
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1answer
202 views

functional determinant

Let $w(x,y)$ be a real-valued symmetric function over the $[0,1]$x$[0,1]$ interval. We also know that if $n$ is an integer, and you pick $n$ values $x_i$ in the $[0,1]$ interval, then the matrix ...
3
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1answer
838 views

Equation of a sphere as the determinant of its variables and sampled points

Searching for an equation to find the center of a sphere given 4 points, one finds that taking the determinant of the four (non-coplanar) points together with the variables $x$, $y$, and $z$ arranged ...
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164 views

Determinants and homomorphisms of general linear groups

Consider the functions $\rho_1:M_1(\mathbb C)\to M_2(\mathbb R)$ where $$\rho_1(a+bi)=\begin{pmatrix} a&b\\ -b&a \end{pmatrix}$$ and $\rho_2:M_2(\mathbb C)\to M_4(\mathbb R)$ where ...