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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3
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2answers
433 views

How to find k given determinant?

So I've got this matrix here, and need to solve for $k$ $$\text{det}\;\begin{pmatrix} 3 & 2 & -1 & 4 \\ 2 & k & 6 & 5 \\ -3& 2 & 1 & 0 \\ 6 & 4 & 2 & ...
3
votes
5answers
138 views

How to find the determinant of this matrix

I have the following matrix: $ \begin{bmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1 \\ 1 & 1 & a & 1 \\ 1 & 1 & 1 & a \\ \end{bmatrix} $ My approach is ...
3
votes
3answers
485 views

How to randomly construct a square full-ranked matrix with low determinant?

How to randomly construct a square (1000*1000) full-ranked matrix with low determinant? I have tried the following method, but it failed. In MATLAB, I just use: n=100; A=randi([0 1], n, n); while ...
3
votes
2answers
806 views

By using properties of determinants show that

$$\begin{vmatrix}1+a^2-b^2&2ab&-2b\\ 2ab&1-a^2+b^2&2a\\ 2b&-2a&1-a^2-b^2\end{vmatrix}=(1+a^2+b^2)^3$$ I have been trying to solve the above determinant. But unfortunately my ...
3
votes
2answers
122 views

Calculating determinant of matrix

I have to calculate the determinant of the following matrix: \begin{pmatrix} a&b&c&d\\b&-a&d&-c\\c&-d&-a&b\\d&c&-b&-a \end{pmatrix} Using following ...
3
votes
3answers
273 views

Minimal polynomial, determinants and invertibility

I need to prove: if a matrix $A$ is invertible, then the minimal polynomial $m_a(0) \neq 0$ There is one definition I am unsure of or need help making more clear. I will proceed with proof by ...
3
votes
2answers
81 views

Calculating $\det(A+I)$ for matrix $A$ defined by products

Let $b_1,\ldots,b_n\in\mathbb{R}$. I have an $n\times n$ matrix $A$ whose entry is given by $a_{ij}=b_ib_j$, and I'd like to show that $\det(A+I)=\sum_{i=1}^nb_i^2+1$. Define $b=(b_1,\ldots,b_n)$. I ...
3
votes
2answers
179 views

Find matrices $X$ such that for any matrix $Y$ we have $\det(X^2 + Y^2) \geq 0$ [duplicate]

What is the characterization of real matrices $X \in \mathbb{R}^{n\times n}$ such that for any real matrix $Y \in \mathbb{R}^{n\times n}$: $$\det(X^2 + Y^2) \geq 0?$$
3
votes
2answers
76 views

Is this determinant bounded?

Let $D_n$ be the determinant of the $n-1$ by $n-1$ matrix such that the main diagonal entries are $3,4,5,\cdots,n+1$ and other entries being $1$. i.e. $$D_n= \det \begin{pmatrix} ...
3
votes
2answers
118 views

When every term of the determinant is zero

The determinant of $A$ is defined as $$ \det(A) = \sum \pm a_{1,i_1} a_{2,i_2} \cdots a_{n,i_n}$$ Suppose $A$ is a real matrix such that every term in the above sum is zero. Is it true that $A$ has a ...
3
votes
2answers
178 views

How do I prove that the following method to find whether a point lies within a polygon is correct?

I came across the following method to determine whether a given point lies inside a convex polygon - however, I'm not sure how to prove it. Given any three points on the plane $(x_0,y_0)$, ...
3
votes
5answers
444 views

How to find Determinant of a matrix

I could not understand the concept while googling. can anybody provide help? what will be the determinant of the following matrix? $$ \left[\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 5 ...
3
votes
2answers
89 views

How find this matrix value of this $\det(A_{ij})$

Find this value $$\det(A_{n\times n})=\begin{vmatrix} 0&a_{1}+a_{2}&a_{1}+a_{3}&\cdots&a_{1}+a_{n}\\ a_{2}+a_{1}&0&a_{2}+a_{3}&\cdots&a_{2}+a_{n}\\ ...
3
votes
4answers
48 views

Find the product of the following determinants:

Find the product of the following determinants: $$\begin{vmatrix} \log_3512 & \log_43 \\ \log_38 & \log_49 \end{vmatrix} * \begin{vmatrix} \log_23 & \log_83 ...
3
votes
3answers
150 views

Is there a way to get all the permutations of $S_4$

I need to calculate the determinant of a $4 \times 4$ matrix by "direct computation", so I thought that means using the formula $$\sum_{\sigma \in S_4} (-1)^{\sigma}a_{1\sigma(1)}\ldots ...
3
votes
2answers
268 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 ...
3
votes
4answers
113 views

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
3
votes
2answers
70 views

how to compute the determinant of a linear map

Let $V$ be the vector space of $m\times n$ matrices over a field $F$. Fix an $m\times m$ matrix $A$ and an $n\times n$ matrix $C$, and consider the map $\phi: V\longrightarrow V$ defined by ...
3
votes
3answers
217 views

Determinant from matrix entirely composed of variables

I don't want the answer, but I'd love to kick in the right direction. I'm really not sure how to approach this question. $$\begin{align} & -6 = det\begin{bmatrix} a & b & c \\ d & e ...
3
votes
2answers
91 views

How to solve this determinant?

I have to solve determinant of the following form: $$a_{ij}=|i-j|+1$$ It looks like this: $$ \begin{pmatrix} 1 & 2 & 3 & 4 & \cdots & n \\ 2 & 1 & 2 & 3 & ...
3
votes
6answers
159 views

Matrix inverse identity

Question: Assuming that all matrix inverses involved below exist, show that $$(\mathbf{A}-\mathbf{B})^{-1}=\mathbf{A}^{-1}+\mathbf{A}^{-1}(\mathbf{B}^{-1}-\mathbf{A}^{-1})^{-1}\mathbf{A}^{-1}$$ in ...
3
votes
5answers
118 views

calculate generally the determinant of $A = a_{ij} = \begin{cases}a & i \neq j \\ 1 & i=j \end{cases}$

calculate generally the determinant of $A = a_{ij} = \begin{cases}a & i \neq j \\ 1 & i=j \end{cases} = \begin{pmatrix} 1 & a & a & · & a \\ · & · & · & · \\ a ...
3
votes
2answers
67 views

Characteristic and minimal polynomial of a special matrix

$H = \begin{bmatrix} 1 & w^{-1} & w^{-2} & ... & w^{1-n}\\ w & 1 & w^{-1} & ... & w^{2-n} \\ w^{2} & w^1 & 1 & ... & w^{3-n} \\ ... & ... & ...
3
votes
3answers
50 views

Matrix set determination

Let $ H=\{ A\in M_2(\mathbb{R}) | A^2=A \},x \in \mathbb{R} $ a) Prove that if $M \in H$ and $\det(M) \neq 0$ then $\det(M)=1$. I tried this using the Hamilton-Cayley relationship, but didn't ...
3
votes
5answers
1k views

Show that the area of a triangle is given by this determinant

This is part of my homework. I'm not sure how to start for this question. Can you guys provide some input/hints? Thank you! Let $A=(x_1,y_2)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in ...
3
votes
3answers
233 views

Rank and determinant of $D$ , an $n\times n$ real matrix, $n\ge 2$

Let $D$ be a $n\times n$ real matrix, $n\ge 2$. Which of the following is valid? $\det(D)=0\Rightarrow \mathrm{rank}(D)=0$ $\det(D)=1\Rightarrow \mathrm{rank}(D)\neq 1$ $\det(D)=1\Rightarrow ...
3
votes
1answer
164 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!
3
votes
1answer
156 views

Is this determinant equal to 1

Let $V$ be a finite dimensional vector space over $\mathbf{C}$ with a hermitian inner product. Let $e=(e_1,\ldots,e_n)^t$ and $f=(f_1,\ldots,f_n)^t$ be orthonormal bases for $V$. There is a matrix ...
3
votes
3answers
47 views

Linear Algebra: Is det({{M,F},{F, M})<0 when det(M)=0?

Suppose that $M$ and $F$ are real matrices. Let $A$ be the block-matrix $$ A= \begin{pmatrix} M & F \\ F & M \end{pmatrix} $$ If $\det(M)=0$ is $\det(A)\leq0$? If not, what conditions need ...
3
votes
2answers
56 views

Is this determinant identity correct?

For complex valued matrices $A,B$ where $B$ is invertible, does $$\det(I+B^{-1}AA^*)=\det(I+AA^*B^{-1})=\det(I+AB^{-1}A^*)=\det(I+A^*B^{-1}A)?$$ Here $A^*$ is the conjugate transform. I guess ...
3
votes
2answers
90 views

Interesting determinant: Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$

Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$, $a_{i,j}=1$ if $i-j\equiv\pm2\pmod n$, and $a_{i,j}=0$ otherwise. Find $\det A$. It seems that the ...
3
votes
3answers
194 views

Determinant of matrix $A^3 + 2A^2 - A - 5I$ Given the eigenvalues of A

So A is a 3 by 3 matrix with eigenvalues -1, 1, 2. And I have to find the determinant of $$A^3 + 2A^2 - A - 5I$$ Let $u$ be the eigenvector for the eigenvalue -1. Let $S = A^3 + 2A^2 - A - 5I$ then ...
3
votes
2answers
863 views

Matrix determinant using Laplace method

I have the following matrix of order four for which I have calculated the determinant using Laplace's method. $$ \begin{bmatrix} 2 & 1 & 3 & 1 \\ 4 & 3 & 1 & 4 \\ -1 ...
3
votes
2answers
49 views

Linear Algebra determinant and rank relation

True or False? If the determinant of a $4 \times 4$ matrix $A$ is $4$ then its rank must be $4$. Is it false or true? My guess is true, because the matrix $A$ is invertible. But there is ...
3
votes
1answer
183 views

Problem with Jacobi's formula for determinants

Jacobi's formula says that: $$\det e^{X}=e^{\operatorname{Tr}(X)}$$ So for any matrix $A$, I could try to find a matrix $X$ (the equivalent to a group generator) such that $A=e^{X}$ holds. But if ...
3
votes
5answers
2k views

For $det(A)=0$, how do we know if A has no solution or infinitely many solutions?

If the determinant det(A) of the matrix A of a non-homogeneous system of equations is 0, then how do we know if it has no solutions or infinitely many solutions? And while we are at it, kindly answer ...
3
votes
2answers
197 views

How to show this determinant $D \not= 0$ (EDIT) maybe figure out is impossible

SORRY, I made a typo. it should be $D \not= 0$,not $D>0$. It is a bit like Vandermonde determinant $$D=$$ $$\begin{vmatrix} 1 & 2 & 3&\cdots &2008&2009 & 2010 & 2011\\ ...
3
votes
1answer
89 views

Characteristic equation for 2-nd order ODE

Given a differential equation $\dot x = Ax$, $x \in \mathbb{R}^n$ we define its characteristic equation as $\chi(\lambda) = \det (\lambda I - A)$. Consider now the second order ODE $$ \ddot x + A x ...
3
votes
1answer
386 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 ...
3
votes
2answers
488 views

Determinant of a special kind of block matrix

I have a $2\times2$ block matrix $M$ defined as follows: $$\begin{pmatrix}X+|X| & X-|X| \\ Y-|Y| & Y+|Y|\end{pmatrix}$$ where $X$ and $Y$ are $n\times n$ matrices and $|X|$ denotes the ...
3
votes
2answers
96 views

A function that looks like determinant

Let $A$ be the $n\times n$ matrix $(a_{ij})$. By Laplace formula, the cofactor expansion along the $j$th row is $$\det(A)=\sum_{j=1}^n (-1)^{i+j}a_{ij}M_{ij}.$$ I'm studying the function ...
3
votes
3answers
47 views

Vandermonde determinant for order 4

I'd like to show the case $n=4$ for the Vandermonde-determinant. It should look like this: $V_4 := \det \begin{pmatrix} 1 & 1 & 1 & 1 \\ x_1 & x_2 & x_3 & x_4 \\ x_1^2 & ...
3
votes
1answer
208 views

block matrices problem

Let $A,B,C$ and $D$ be n by n matrics such that $AC=CA$. Prove that $\det \begin{pmatrix} A & B\\ C & D \end{pmatrix}=\det(AD-CB)$. The solution is to first assume that $A$ is invertible and ...
3
votes
2answers
69 views

Factorization of a linear combination of matrices

I'm trying to understand the determinant from Axler Sheldon's paper and there is one point in the very beginning that I don't understand :S (Link below to the paper) ...
3
votes
1answer
112 views

A variation of Cauchy's determinant

Prove the following identity: $$\det_{_{1\leq i,j \leq n}}\left(\frac{1}{(x_i+y_j)^2}\right)=\det_{_{1\leq i,j \leq n}}\left(\frac{1}{x_i+y_j}\right)\text{perm}_{_{1\leq i,j \leq ...
3
votes
2answers
103 views

Calculating the determinant of this complicated matrix

I am calculating the characteristic polynomial for this matrix: $$A = \begin{pmatrix} 1 & 2 & \cdots & n \\ 1 & 2 & \cdots & n \\ \vdots & \vdots & \cdots & \vdots ...
3
votes
1answer
3k views

Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory

In order to prove the following identity: $$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ Instead of checking this by brute force, Landau writes thr product of ...
3
votes
1answer
79 views

Why Vandermonde's determinant divides such determinant?

Assume that $$ W(x_1,...,x_n;k)=\left [ \begin{array}{rrrrrrrr} 1 & x_1 &... & x_1^{n-2} & x_1^k \\ 1 & x_2 &... & x_2^{n-2} & x_k \\ & & \ddots \\ 1 & ...
3
votes
2answers
222 views

Parallelogram area using determinant

Given a Parallelogram with the co-ordinates: $(a+c, b+d), (c,d), (a, b)$ and $(0, 0)$ I have to prove that the area of the Parallelogram is: $|ad-bc|$ as in the determinant of: $$\begin{bmatrix} a ...
3
votes
1answer
93 views

eigenvalues of block matrix with the eigenvalues of one block already known

Give a matrix which can be decomposed into 4 parts $B = \left[\begin{matrix}A &I \\ -I &0\end{matrix}\right]$ where $I$ denotes the identity matrix and $0$ is a zero matrix. It's easy to ...