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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0
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1answer
31 views

Differentiation Involving Determinant.

I have to compute the following differentiation : $$\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times ...
-1
votes
1answer
46 views

How prove this determinant can't zero

Let $x,y,z\neq 0$ be real numbers, show that $$f(x,y,z)=\begin{vmatrix} \sqrt{x^2+y^2}&|x|&|y|\\ |y|&\sqrt{y^2+z^2}&|z|\\ |x|&|z|&\sqrt{x^2+z^2} \end{vmatrix}\neq 0$$ or it ...
3
votes
2answers
131 views

Determinant of specially structured block matrix

How do you compute the determinant of the following $nd \times nd$ block matrix? $$M = \begin{bmatrix}A+B & A & A & \dots & A & A\\ A & A+B & A & \dots & A & ...
14
votes
7answers
1k views

Determinant of a specially structured matrix ($a$'s on the diagonal, all other entries equal to $b$)

I have the following $n\times n$ matrix: $$A=\begin{bmatrix} a & b & \ldots & b\\ b & a & \ldots & b\\ \vdots & \vdots & \ddots & \vdots\\ b & b & \ldots ...
1
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2answers
3k views

Leading principal minors

How many leading principle minors are there for a 4X4 matrix? please explain in detail. I know for a 3X3 matrix.
3
votes
2answers
400 views

Finding characteristic polynomial of adjacency matrix

Short question im having a tad difficulty with. I'm trying to find the characteristic polynomial of a graph that is just a circle with n vertices and n edges. I think the adjacency matrix should ...
2
votes
2answers
32 views

How do I find value of a and b in this matrix question?

This is a question from a homework sheet my teacher gave. I already did alternate a. Alternate b is quite confusing! It asks to find the value for a and b. I don't really know what to do but here's ...
0
votes
1answer
47 views

Calculation of characteristic polynomial

I have to determine the characteristic polynomial of the matrix $$A = \begin{pmatrix} 0 & 0 &\cdots &0& -a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots ...
3
votes
1answer
638 views

Finding the eigenvalues of a $3N \times 3N$ block matrix

I have a block matrix of size $3N \times 3N$ of the form $$B = \begin{bmatrix} A & C & \ldots & C\\ C & A & \ldots & C\\ \vdots & \vdots & \ddots & \vdots\\ C ...
4
votes
1answer
68 views

Calculating the determinant of a matrix using its rank

Let A, B, C and D be real n×n matrices. If $$\operatorname{rank} \begin{bmatrix} \ A & B \\[0.3em] \ C & D \\[0.3em] \end{bmatrix} = n$$ then show that $$\det ...
4
votes
1answer
453 views

determinant recursive formula of a specific matrix

For a field $K, n \in \mathbb{N}_{>0}$ and $\lambda \in K$ let $A_{n, \lambda} \in \textrm{Mat} (n,K) $ be the following matrix with entries $\lambda$ on the diagonal, $-1$ on both minor diagonals ...
1
vote
1answer
48 views

Adjoint of an adjoint of a matrix

Can you please help me on this question? $\DeclareMathOperator{\adj}{adj}$ $A$ is a real $n \times n$ matrix; show that: $\adj(\adj(A)) = (\det A)^{n-2}A$ I don't know which of the expressions ...
0
votes
0answers
18 views

Inverse Gramian matrix

Show that inverse Gramian matrix is Gramian matrix Here is my idea. $\Gamma ^ {-1} = \frac{A}{|\Gamma|}$, where A is transposed matrix of cofactors (not sure about the term, correct me please), ...
2
votes
2answers
884 views

The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
2
votes
2answers
575 views

Gram Determinant equals volume?

I have been trying to solve this problem of finding the 'n-volume' of a paralleletope spanned by m vectors, where clearly m =< n. In general, for computational purposes, what I have managed to do ...
4
votes
1answer
32 views

Matrix consisting of cosines of differences

Consider the following matrix: $$\left[\begin{array}{cccc} \cos(x_1-y_1) & \cos(x_1-y_2) & \ldots & \cos(x_1-y_n) \\ \cos(x_2-y_1) & \cos(x_2-y_2) & \ldots & \cos(x_2-y_n) \\ ...
3
votes
3answers
39 views

Matrix with a certain pattern

Consider the following matrix: $$\left[\begin{array}{cccc} 1+x_1y_1 & 1+x_1y_2 & \ldots & 1+x_1y_n \\ 1+x_2y_1 & 1+x_2y_2 & \ldots & 1+x_2y_n \\ 1+x_3y_1 & 1+x_3y_2 & ...
1
vote
1answer
89 views

Determinant of augmented matrices.

Let $A$ and $B$ be $n \times n$ real matrices. How can I show that $\det \begin{bmatrix} A & B \\[0.3em] -B & A \\[0.3em] \end{bmatrix} \geq 0 $?
4
votes
1answer
58 views

Inverting an $n \times n$ matrix using determinant

We're asked to invert the following matrix with the help of guided questions. $$\begin{pmatrix} 1 + a_1 & 1 & \cdots & 1 \\ 1 & 1+a_2 & \ddots & \vdots \\ \vdots & \ddots ...
7
votes
1answer
202 views

Determinant of $2\times 2$ Block Matrix

I would like to know the proof for: The determinant of the block matrix\begin{pmatrix} A & B\\ C& D\end{pmatrix} equals $(D-1) \det(A) + \det(A-BC) = (D+1) \det(A) - \det(A+BC),$ when $A$ ...
-7
votes
1answer
112 views

Generalized determinant of order $n+2$

How to solve following determinant of order $n+2$ to get eigenvalues? $\begin{vmatrix} -\lambda & 0 & 1 & 1 & 1 & \cdots & 1 \\ \dfrac{n}{\lambda} & ...
7
votes
2answers
262 views

Non-negative determinant of a block matrix

Here's the problem I've been stuck on for some time now. Let $A,B \in M_n(\mathbb{R})$. Let $C= \begin{bmatrix} A & B \\ -B & A \\ \end{bmatrix} $ be a real ...
1
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0answers
103 views

Prove that $ \det{\begin{bmatrix}A & B \\-B & A\end{bmatrix}}\geq 0$ [duplicate]

Let $A,B \in M_n(\mathbb{R})$. Prove that $\det{\begin{bmatrix}A & B \\-B & A\end{bmatrix}}\geq 0$. I know that there is a theorem which says that if $E,F,G,H \in M_n(\mathbb{F})$ and ...
0
votes
1answer
14 views

For integers $n>1$ , $k$ , does there exist matrix $A$ with integer entries and first row $(1,2,…,n)$ such that $\det A=k$?

Let $n >1$ be an integer , then is it true that for any integer $k$ , there exist a matrix $A \in M(n,\mathbb Z)$ with first row of $A$ as $(1,2,...,n)$ such that $\det A=k$ ?
10
votes
2answers
3k views

Coefficients of characteristic polynomial of a matrix

For a given matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed from the columns and rows with indices from $J$. If the characteristic polynomial of A is ...
4
votes
2answers
2k views

Block matrix determinant

I have encountered an statement several times while proving determinant of a block matrix. $$\det\pmatrix{A&0\\0&D}\; = \det(A)\det(D)$$ where $A$ is $k\times k$ and $D$ is $n\times n$ ...
5
votes
4answers
144 views

Products of adjugate matrices

Let $S$ and $A$ be a symmetric and a skew-symmetric $n \times n$ matrix over $\mathbb{R}$, respectively. When calculating (numerically) the product $S^{-1} A S^{-1}$ I keep getting the factor $\det S$ ...
1
vote
1answer
43 views

Derivatives using matrices good

$$\left|\begin{matrix} (1+x)^{a_1b_1} & (1+x)^{a_1b_2} & (1+x)^{a_1b_3} \\ (1+x)^{a_2b_1} & (1+x)^{a_2b_2} & (1+x)^{a_2b_3} \\ (1+x)^{a_3b_1} & (1+x)^{a_3b_2} & (1+x)^{a_3b_3} ...
0
votes
2answers
171 views

Oddity: Determinant of Skew-Symmetric $n\times n$ Matrices [closed]

Let $A \in M_{n×n}(\mathbb{R})$ be a skew-symmetric matrix, i.e., $A^t = −A$. Prove that if $n$ is odd, then $\det{A} = 0$.
0
votes
1answer
33 views

For what values of $x_1, x_2, x_3, x_4$ is the matrix $A$ invertible? [duplicate]

For what values of $x_1, x_2, x_3, x_4$ is the matrix $A=\begin{pmatrix}1 & 1 & 1 &1 \\ x_1 & x_2 & x_3 &x_4 \\ x_1^2& x_2^2 & x_3^2 & x_4^2\\ x_1^3& x_2^3 ...
1
vote
1answer
33 views

The property that det(A) = prod of A's eigenvalues, and tr(A) = sum of A's eigenvalues

Do these two properties fail to be true, if A's characteristic polynomial fails to split? If so, then do we usually work in a vector space with the ground field = $\mathbb{C}$, when we want to use ...
2
votes
0answers
21 views

Show that matrix is totally unimodular

I want to show that this matrix is totally unimodular: \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & ...
7
votes
1answer
2k views

Proving determinant product rule combinatorially

One of definitions of the determinant is: $\det ({\mathbf C}) =\sum_{\lambda \in S_n} ({\operatorname {sgn} ({\lambda}) \prod_{k=1}^n C_{k \lambda ({k})}})$ I want to prove from this that ...
1
vote
1answer
30 views

determinants of matrix with adjoints of order 2

Let $A$ be a square matrix of order $2$ with $\lvert A \rvert\ne 0$ such that $\big\lvert A+\lvert A \rvert \operatorname{adj} (A)\big\rvert=0$, then the value of $$\big\lvert A-\lvert A \rvert ...
1
vote
1answer
35 views

Rank of square matrix $A$ with $a_{ij}=\lambda_j^{p_i}$, where $p_i$ is an increasing sequence

Let $$ A = \begin{bmatrix} \lambda_1^{p_1} & \lambda_2^{p_1} & \cdots & \lambda_n^{p_1} \\ \lambda_1^{p_2} & \lambda_2^{p_2} & \cdots & \lambda_n^{p_2} \\ ...
4
votes
2answers
49 views

When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$

Let 'A' be a $2 \times 3$ matrix where as B be a $3 \times 2$ matrix if $\det(AB) = 4$ the find value of the $\det(BA)$ My attempt: I took A = $$ \begin{bmatrix} 2 & 0 &0\\ ...
0
votes
0answers
17 views

Coefficient of bivariate polynomial as a determinant of matrix

Given $$ \begin{bmatrix} a\\ b\\ c\\ d\\ \end{bmatrix}=\begin{bmatrix} a_0t^3+a_1st^2+a_2s^2t+a_3s^3\\ a_4t^2+a_5st+a_6s^2\\ a_7t+a_8s\\ a_9\\ \end{bmatrix} $$ the following equation holds: $$ ...
1
vote
0answers
47 views

Given matrices $A,B, \det(A) = 2$ find $\det(B)$

Given matrices $A$,$B$, $\det(A) = 2$ Find $\det(B)$ $$A= \begin{pmatrix} a & 1 & b \\ 2 & 3a & 1 \\ b & 1 & 2a \\ \end{pmatrix}, ...
7
votes
1answer
623 views

Determinant of Matrix with uncomputable values.

Calculate the determinant of the matrix $$ \begin{pmatrix} 10^{10} & 10^{10^{10}} & 11^{11^{11}} & 1 & 0 \\ 2^{2^2} & 3^{3^3} & 7^{7^7} & 0 & 1 \\ 11 & ...
1
vote
2answers
27 views

Proving $\det \big(Df\big|_x\big)=0$ for a function into unit circle

Let $f:\mathbb{R}^2\to S$ where $S=\{x\in\mathbb{R}^2:\, ||x||=1\}$. Prove that $\det \big(Df\big|_x\big)=0$ for all $x$. I'm having trouble attacking this. So I need to show that there is some ...
3
votes
3answers
107 views

Prove that $\det(A^{T}A) \neq 0$

How to prove that $\det(A^{T}A) \neq 0$ if coloumns of $A$ are linearly independent, without using Cauchy-Binet formula? $A$ is real matrix.
0
votes
0answers
9 views

Proof attempt: A is an antisymmetric matrix (of even size). B is another matrix such that $b_{i,j}=a_{i,j}+c$. Prove that |A|=|B| [duplicate]

I asked this question but all the answers I got were outside of my scope of understanding, so here is as close as I got to a solution: $$\begin{bmatrix} c & a_{12}+c &...&&a_{1n}+c \\ ...
1
vote
1answer
23 views

tell Positive Definite Matrices by the sign of determinants

This is from my textbook I don't understand why it didn't mention the other situation which is $det(A_k) <0 $ for all k, and we stall have positive pivot because ...
1
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1answer
46 views

Finding determinant of matrix through row operations [problem help]?

I am having trouble understanding a problem that my Linear Algebra class gave. I understand that determinants can be found through row operations with the following points: 1.) Adding a multiple ...
0
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0answers
37 views

determinant of a matrix with increased off-diagonal elements

I have symmetric matrix which is formed by complex integer vectors as follows \begin{bmatrix} \|f_1\|^2 & af_2^Hf_1 & bf_3^Hf_1 & \dots \\ a^*f_1^Hf_2 & \|f_2\|^2 & cf_3^Hf_2 & ...
2
votes
0answers
45 views

Determinants Proof Can't Solve

Let $v_1,\cdots ,v_n$ be vectors in $\mathbb R^n$. Define $w_i$ as the vector in $\mathbb R^{n+1}$, which is just $v_i$ with a $0$ added to the beginning. (So if $v_i = (1,0,1)$, then $w_i = ...
8
votes
2answers
98 views

Determinant of Tridiagonal matrix

I'm a bit confused with this determinant. We have the determinant $$\Delta_n=\left\vert\begin{matrix} 5&3&0&\cdots&\cdots&0\\ 2&5&3&\ddots& &\vdots\\ ...
30
votes
9answers
30k views

How to show that $\det(AB) =\det(A)\det(B)$

Given two square matrices $A$ and $B$, how do you show that $\det(AB) = \det(A)\det(B)$, where $\det(\cdot)$ is the determinant of the matrix?
6
votes
3answers
111 views

A is an antisymmetric matrix (of even size). B is another matrix such that $b_{i,j}=a_{i,j}+c$. Prove that |A|=|B|

I know that B would look something like this: $$\begin{bmatrix} c & a_{12}+c &...&&a_{1n}+c \\ -a_{12}+c & c &...&&a_{2n}+c \\ . \\ . \\ . \\ -a_{1n}+c & ...
0
votes
1answer
21 views

A family of vectors is linearly independent.

Let $K$ be a field and $E$ be a $K$-vector space of dimension $n$. Let $\phi$ be an endomorphism of $E$. Let $(\lambda_1,\cdots,\lambda_n)$ be a family of distinct scalars and $(x_1,\cdots,x_n)$ be ...