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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2answers
63 views

prove: $\left(A^*\right)^*=\left|A\right|^{n-2}A$ [duplicate]

Suppose square matrix $A$ with order-n, and $A^*$ is it's adjugate matrix, when $n>2$, prove: $\left(A^*\right)^*=\left|A\right|^{n-2}A$ Proof: when $A$ is invertible, ...
0
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2answers
196 views

Possible matrix-determinant identity

Is $$\det(I+ ABB^*A^*C^{-1})=\det(I+ B^*A^*ABC^{-1})$$ where $I$ is identity matrix, $A,B,C$ are complex valued matrices. And $C$ is $(I+X)$ where $X$ is PSD. I know that this makes $ABB^*A^*$ and ...
1
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1answer
35 views

A possible determinant identity

Is $$\det(I+A(I+B)^{-1})=\det(I+A^*(I+B)^{-1})$$ where $I$ is identity matrix, $A,B$ are positive semi-definite complex valued matrices and $A^*$ is the conjugate (Hermitian) transpose of $A$. ...
2
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2answers
2k views

The determinant of adjugate matrix

I have the following proof that I would like to be walked through because I'm not intuitively seeing what to do: If $A$ is $n\times n$, prove $\det\left(\operatorname{adj}(A)\right) = \det(A)^{n-1}$. ...
3
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1answer
89 views

solving the $ \varepsilon_{ijk}\varepsilon_{lmn}$(Levi Civita)

How can I solve this : $ \varepsilon_{ijk}\varepsilon_{lmn}=??$ I know that It can be solve with 2 determinants but I don't know how.and I don't what are the determinants!
2
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2answers
117 views

The relation between a real matrix and a complex matrix. [closed]

Let $A$ and $B$ be $n\times n$ real matrices. Show that $$\det\begin{pmatrix}A & -B \\ B & A \end{pmatrix}=|\det(A+iB)|^{2}.$$
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0answers
48 views

inequalities for the relation between $\det(A+A')$ and $\det(A)$, where $A$ is positive definite (not necessarily symmetric)

I was wondering if there is any relationship (in the form of inequality) between $\det(A+A')$ and $\det(A)$, where $A$ is a positive definite but not necessarily symmetric.
2
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0answers
48 views

Reference request on pseudo-determinants

I am looking for a reference on pseudo-determinants$^{(1)}$. I am mostly interested on general and/or basic equalities and properties such as those obtained for determinants. Any pointers would be ...
0
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2answers
35 views

Calculating determinants. Help appreciated

Does anyone know how I would go about answering this question? Any feedback is appreciated. I'm not too sure where to start. (a) Calculate the determinant of $D = \begin{bmatrix} 1 & 2\\ 2 ...
2
votes
3answers
125 views

Relation between $A^{-1}$ and $\det A$

I know there is a relationship between an $n\times n$ matrix $A^{-1}$ and $(\det A)^{-1}$. That is, $A^{-1}$ is equal to $(\det A)^{-1}$ times what? How to use a formula to express the relationship ...
2
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2answers
122 views

is it true that $\det(I+A)>0$ , if $\det(A)>0$?

I saw an inequality for $n\times n$ matrices. I was wondering if the inequality is true or not? Does $\det(A)>0$ imply $\det(I+A)>0$?
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3answers
1k views

Intuition behind Matrix being invertible iff determinant is non-zero

I have been wondering about this question since I was in school. How can one number tell so much about the whole matrix being invertible or not? I know the proof of this statement now. But I would ...
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votes
1answer
360 views

$dxdydz \to -r^2\sin(\theta)\sin(\phi+\theta)dr d\phi d\theta$?

So I got this answer $-r^2\sin \theta\sin(\phi+\theta)dr d(\phi)d(\theta)$ which I think is wrong because I googled it and it must be $-r^2\sin\theta dr d\phi d\theta,$ but $\sin(\phi+\theta$) clearly ...
4
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0answers
78 views

The determinant of a special matrix

Recently, I encounter the problem of calculating the determinant of the following matrix $$\left(\begin{array}{cccc} \sin(\theta_1) & \sin(\theta_1 + \delta_1) & \cdots & \sin(\theta_1 + ...
6
votes
1answer
143 views

Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
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2answers
715 views

Finding the determinant of a $4\times4$ matrix

How does one find the determinant of a $4\times 4$ matrix? I am using Cramer's rule to solve a system of linear equations but don't know how to find the determinant of a $4\times 4$ matrix. Our matrix ...
1
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1answer
55 views

On determinants computation

How can be proved this identity between determinants? $$\left|\begin{array}{cccc} 1&a&c&ac\\ 1&b&c&bc\\ 1&a&d&ad\\ 1&b&d&bd \end{array}\right|=\left| ...
4
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2answers
114 views

Determinants of Block Matricies

I read on wikipedia that $Det \begin{pmatrix} A & B\\B& A\end{pmatrix}$ is equal to $ Det(A+B)Det(A-B) $ if $A$ and $B$ commute. Does this hold true even if $ A $ and $ B$ are not ...
2
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0answers
95 views

The most efficient algorithm to solve the following problem

Is there an efficient optimization algorithm to solve the following problem? $(\alpha,\beta,\gamma,\cdots) =$ argmax $\sum_{i}\log(\alpha a_i+\beta b_i+\gamma c_i+\cdots)$, s.t. ...
4
votes
1answer
128 views

Prove that $\det A = 1$ with $A^T M A = M$ and $M = \begin{bmatrix} 0 & I \\ -I &0 \end{bmatrix}$. [duplicate]

Prove that $\det A = 1$ with $A^T M A = M$ and $M = \begin{bmatrix} 0 & I \\ -I &0 \end{bmatrix}$ ($I$ is the identity matrix of order n).
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2answers
72 views

Help with calculating the determinant

Does anyone know how to go about answering the following? Any help is appreciated! Calculate the determinant of $D = \begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}$ and use it to find ...
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2answers
218 views

Why is the determinant zero iff thee column vectors are linearly dependent?

I see a lot of references to this all over the web, but I can't find an actual explanation for this anywhere.
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1answer
509 views

Simultaneous Equations That Should Be Inconsistent Has a Unique Solution

Find the values of $k$ for which the simultaneous equations do not have a unique solution for $x, y$ and $z$. Also show that when $k = -2$ the equations are inconsistent $$kx + 2y +z =0$$ $$3x + 0y ...
2
votes
1answer
315 views

Determinant of matrix obtained by commuting matrices

The Question is to prove that : For Commuting $n\times n$ matrices $A,B,C,D$ over a field $F$, Determinant of $\left(\begin{array}{cccc} A & B \\ C & D \\ \end{array} \right)$ is given by ...
0
votes
2answers
262 views

Given a matrix factored into a product, how do you determine the determinant?

I'm preseneted with the question: Suppose that a 3x3 matrix A factors into the product of the two matrices below: \begin{matrix} 1 & 0 & 0 \\ I21 & 1 & 0 \\ I32 & I32 & ...
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3answers
71 views

Expressing a $3\times 3$ determinant as the product of four factors

I am attempting to express the determinant below as a product of four linear factors $$\begin{vmatrix} a & bc & b+c\\ b & ca & c+a\\ c & ab & a+b\\ \end{vmatrix} = ...
0
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0answers
77 views

Linear dependance and Wronskian determinant

I am asked to show that the fuctions $e^x, \cos(x) \text{ and } x^2$ are linearly independent. I wanted to use the Wronskian determinant in order to prove the above property. We have: $$W= ...
0
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0answers
68 views

How is determinant defined by variable elimination?

I've been chasing origins of determinants for quite a while now, and having depleted all literature I have access to, I'm trying to find some hints here. Many books describe determinants as solutions ...
0
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3answers
52 views

Factorising a 3 x 3 determinant - What Am I doing Wrong?

$$\begin{vmatrix} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \\ \end{vmatrix}$$ subtracting the top row from the middle and bottom rows $$ = \begin{vmatrix} 1 & a & a^3 ...
4
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3answers
124 views

Calculating determinant with real number on diagonal and units everywhere else

I'm solving a problem and I'm having difficulties in calculation of the determinants of two matrices. There is two $N\times N$ matrices: $$\left( \begin{array}{cccc} a & 1 & \ldots ...
1
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1answer
134 views

simpler way to calculate a determinant?

Simpler way to calculate this? $$A = \begin{bmatrix}\lambda -2 & 2 & 0 \\ 2 & \lambda -1 & 2 \\ 0 & 2 & \lambda \end{bmatrix}$$ my method: \begin{align*} \det A &= \det ...
0
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1answer
246 views

Step in Euler's rotation theorem

I have been examining the matrix proof for Euler's rotation theorem on Wikipedia. I have deduced every step up to proving that $\det (R - I) = 0$ for any rotation matrix R. However, I'm having ...
0
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1answer
73 views

Calculate the determinant? any hints

Calculate the determinant $$\begin{vmatrix} n & s_1 & s_2 & \cdots & s_{n-1} \\ s_1 & s_2 & s_3 & \cdots & s_n \\ s_2 & s_3 & s_4 & \cdots & s_{n+1} \\ ...
1
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4answers
197 views

Calculate the Determinant?

$$D=\begin{bmatrix} 246 & 427 & 327 \\ 1014 & 543 & 443 \\ -342 & 721 & 621 \\ \end{bmatrix}$$ What's the trick? Hints? Of course I know calculate by definition... Please ...
7
votes
2answers
194 views

Historical meaning and usage of determinant

Can anyone please explain how, why, and where determinants were developed/formalized? What was their historical usage? Why were they initially formulated and what were they used for (and later ...
3
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2answers
186 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 $Y \in \mathbb{R}^{n\times n}$: $$\det(X^2 + Y^2) \geq 0?$$
4
votes
3answers
303 views

A basic question on determinant and rank of a matrix

How to prove that if the determinant of a $n \times n$ matrix is zero then the rank is less than $n$. I can prove the converse. Only a hint is enough. My definition of rank is the maximum number of ...
3
votes
1answer
290 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 ...
0
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1answer
37 views

Determinant of elementary matrix of type 2

I would like to have an induction proof of why the determinant of an elementary matrix with two rows swapped equals -1. I'm brand new to determinants. Thanks.
2
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2answers
272 views

Determinant of permutation matrix (elementary matrix of type 2)

I would like to know why the determinant of a permutation matrix of size nxn (elementary matrix of size nxn of type 2) is -1. I'm brand new to determinants and I've tried expanding it and using ...
1
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0answers
260 views

How to determine that a certain eigenvalue is doubly degenerate?

Given a symmetric matrix $X$. I ask myself, how to determine that a certain eigenvalue $\lambda$ is (exactly) doubly degenerate? I thought about several approaches: Calculate the derivative of $ ...
2
votes
2answers
723 views

Calculate the determinant of a matrix multiplied by itself confirmation

If $ \det B = 4$ is then is $ \det(B^{10}) = 4^{10}$? Does that also mean that $\det(B^{-2}) = \frac{1}{\det(B)^2} $ Or do I have this completely wrong?
7
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5answers
478 views

Determine the value of a second determinant based on the first

I know the theory of determinants, but I have no idea how to apply it to this problem. Suppose $$\det\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i \end{bmatrix} = 6$$ What is the value ...
2
votes
2answers
420 views

Prove that invertible metrices set is an open set in a given space, and the determinant is continuous [duplicate]

Given a matrix $M_{n\times m}$, we can think about it as a vector in $\mathbb{R}^{n\times m}$ (How come?). How can I prove that the set of all the invertible metrices of size $n\times n$ is an open ...
4
votes
1answer
105 views

Calculate the determinant

Calculate the determinant $$\begin{align*}D[n]=\begin{array}{cccccc} b & b & b & \dots & b & a \\ b & b & b & \dots & a & b \\ \vdots & \vdots & ...
0
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3answers
921 views

Solving for unknown value using properties of determinant

Problem : If $ax^4 +bx^3+cx^2+dx+e= $ $$ \begin{vmatrix} x^3+3x & x-1 & x+3 \\ x+1 & -2x & x-4 \\ x-3 & x+4 & 3x \\ \end{vmatrix} $$ ...
1
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4answers
127 views

Calculate the determinant of the matrix $(a_{ij})$ where $a_{ij}=a+b$ when $i=j$, and $a_{ij}=a$ otherwise

The matrix is $n\times n$ , defined as the following: $$ a_{ij}=\begin{cases} a+b & \text{ when } i=j,\\ a & \text{ when } i \ne j \end{cases}. $$ When I calculated it I got the ...
1
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1answer
96 views

Prove that the determinant of polynomials is zero

Prove that this determinant is zero (this matrix is $n\times n$): $$\begin{vmatrix} f_1(a_1) & f_1(a_2) & \cdots & f_1(a_n) \\ f_2(a_1) & f_2(a_2) & \cdots & f_2(a_n) \\ \vdots ...
3
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1answer
569 views

On the difference of two positive semi-definite matrices

I am relatively new to linear algebra, and have been struggling with a problem for a few days now. Say we have two positive semi-definite matrices $A$ and $B$, and further assume that $A$ and $B$ are ...
5
votes
4answers
2k views

$\det(A + B) = \det(A) + \det(B)$?

Well considering two $n \times n$ matrices does the following hold true: $$\det(A+B) = \det(A) + \det(B)$$ Can there be said anything about $\det(A+B)$? If $A/B$ are symmetric (or maybe even of the ...