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
52 views

Determinants of $3\times3$ matrices with full rank

I have two $3\times3$ matrices $A$ and $B$ where $$A = [c_1 : c_2 : c_3]$$ $$B = [c_1 : c_1 + c_2 : c_1+c_2+c_3]$$ where $c_i$ is the $i^{th}$ column of $A$. Given that $|A| = 1$, I am to find the ...
0
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1answer
43 views

$2X2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors

Give an example of $2X2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors. I would like to know a systematic answer of how to get this. My guess ...
3
votes
3answers
268 views

For which $x$ is the determinant 0?

For which values of $x \in \mathbb{R}$ does the determinant of the matrix $$ M = \begin{pmatrix} x & 0 & 1 & 2 \\ 2 & x & 0 & 1 \\ 1 & 2 & x & 0 \\ 0 & 1 ...
0
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0answers
44 views

matrix determinant changes when doing row operation, so weird O_o

To calculate the determinant of a matrix, you can subtract a row by another, and the determinant will not change. However, in the following matrix, the determinant is -2. \begin{bmatrix} 1 ...
2
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0answers
53 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
1
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1answer
61 views

Cramers Rule. The why and how.

Can someone explain how Cramer's rule works. I understand the mechanics of it, and it's fairly straightforward to show algebraically that it's equivalent to GJ and substitution, but what's happening ...
0
votes
2answers
64 views

Fields over which a matrix is not invertible

I am trying to find the fields over which the matrix: $\left(\begin{matrix} 1 & 2 & 3 \\ 0 & -1 & 2 \\ 1 & 0 & -2 \end{matrix}\right) $ is not invertible. I have ...
-1
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4answers
62 views

If $A$ is a $3 \times 3$ matrix and $\det(A) = 4$, then compute $\det(((-9A)^4)^T)$. [closed]

Given a $3\times3$ matrix $A$ $$A= \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} $$ and $\det(A)=4$ Calculate $\det(((-9)\cdot A)^4)^T$.
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2answers
24 views

If rank of $(m+1)\times n$ matrix is $m+1$, then some $(m+1)\times (m+1)$ submatrix has non-zero determinant.

I can't understand this : If I have a $(m+1)\times n$ matrix and if its rank is $m+1$, then some $(m+1)\times (m+1)$ submatrix has non-zero determinant. How is it so?... kindly help.
4
votes
1answer
54 views

Determinant of a $n\times n $ matrix

Let $n$ be a positive odd integer and let $A$ be a symmetric $n\times n$ matrix of integer entries such that $a_{ii}=0,i=1,2.....n$. Show that the determinant of $A$ is even. I tried using ...
2
votes
1answer
34 views

Writing the scalar product using a determinant

Let $A \in \mathbb{R}^{n \times n}$ be symmetrical and positive definite. Does the following statement hold true for $x \in \mathbb{R}^n$? $$\det(x^TAx) = \det(x^TxA)$$ And if so, how can it be ...
0
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0answers
25 views

Derive the determinant of circulant

Let $$ \sigma\in S_n $$ denote the permutation given by $$\sigma\in \begin{pmatrix} 1 & 2 & 3 & ...& n\\ n & 1 & 2 & ... & n-1\\ \end{pmatrix} $$ and let $$ P = ...
0
votes
1answer
23 views

xA=0 sufficient condition for zero determinant?

Let A be a symmetric n by n matrix and x be a 1 by n vector. If I find one x such that xA=0, does it mean A is singular?
0
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1answer
47 views

If A is some invertible $n \times n$ matrix then show $\det(A^n) = (\det(A))^n$ for all $n\in \mathbb{Z}$

So there exists $A^{-1}$. I am assuming $\det(AB)=\det(A)\cdot\det(B)$ and $(A^d)^f=(A^{df})$ I know the proof for $\det(A^{-1})=(\det(A))^{-1}$ is: $\det(I_n)=1$ $\det(A\cdot A^{-1})=1$ ...
1
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2answers
55 views

Determinant matrix proof

Let $A$ be an $n\times n$ matrix and $i,j,k$ be $1\leq i,j,k\leq n$ and $\alpha,\beta \in \mathbb{R}$. I am supposing that $\bf{a}_k$(the $k$-th row) is equal to $\alpha \bf{a}_i+\beta \bf{a}_j$. ...
0
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0answers
40 views

Prove statement about cofactor.

Let $A$ be a $n$ x $n$ matrix $\in R$ and $det(A)=2$ , prove that atleast one of its cofactors is odd.
3
votes
1answer
37 views

Symmetric groups and matrices

I am currently working through this question. I have completed part (a) and (c), however I am unable to make any progress with (b). I know $S_n$ is the symmetric group on n symbols, and that it has ...
0
votes
1answer
19 views

Express m-th times switched rows matrix A in terms of determinant A and m

Let $A'$ be obtained from the square matrix $A$ by interchanging pairs of rows (columns) m times. Express $\det A'$ in terms of $\det A$ and m. I have this question in my Assignment, but I unable to ...
0
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1answer
20 views

Find Determinant of A, when the Product of A and Transpose of A is Identity

If $A^T . A = I$, prove that determinant A = +-1. I don't even know where to start. Can somebody please give me a good start at least.
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2answers
30 views

A limit-determinant question

Interesting question, I don't know where to start. I dont really know how to use this format, so I PrtScr the question.
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2answers
38 views

Confusions about Linear Algebra (determinants) [closed]

So I have been taught how to find determinants if given a size nxn matrix. I know how to do it, but I seriously do not understand why it would work! Even for the simplest determinant of a 2x2 matrix, ...
1
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3answers
54 views

Determinant as a number that tells if a system has solution or not

There are many ways to define and interpret determinants. The one I'm more interested right now is the one that better describes its name: a number that can determinate if a system of linear equations ...
0
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0answers
24 views

determinant of the covariance matrix of a normal distribution

Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol ...
3
votes
2answers
49 views

Determinants of 'block' matrices

I am trying to simplify the determinant of \begin{pmatrix}C&A\\B&0\end{pmatrix} where $A$ and $B$ are square $m\times m$ and $n\times n$ matrices, and $C$ is some $m\times n$ matrix, $0$ is ...
1
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1answer
42 views

Proving the Determinant of a Tridiagonal Matrix

Let $A_n$ denote an $n \times n$ tridiagonal matrix. $$A_n=\begin{pmatrix}2 & -1 & & & 0 \\ -1 & 2 & -1 & & \\ & \ddots & \ddots & \ddots & \\ & ...
0
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1answer
46 views

The determinate of a matrix

The matrix $$\left[\begin{array}{ccc} 30&20&30\\ 40&50&20\\ 30&30&20 \end{array}\right]$$ I tried solving it for myself and got $12000$, but math way tells me its $-1000$. ...
0
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2answers
47 views

Prove statement about determinants.

$A$ is a $3\times 3$ matrix over $\mathbb{R}$, I want to show that if $$\det(A + I_3)=\det(A+2I_3),$$ then $$2\det(A+I_3) + \det(A-I_3) + 6 = 3\det A.$$ Can you help me?
0
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2answers
65 views

Does a matrix $A$ need to have $\det A \neq 0$ to even have a rank?

Does a matrix $A$ need to have $\det A \neq 0$ to even have a rank? So I've had this uneasy feeling that the rank could not be calculated for a matrix $4\times 4$ which had two identical columns, and ...
1
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1answer
36 views

Using Cramer's rule, solve the following.

$$x + y + z = 6$$ $$3x - y + 2z = 7$$ $$ 3y -4z = -6$$ Tried everything. When I check my answer its incorrect, even when I check the example in my handbook I see its answer is wrong. Would like ...
3
votes
2answers
80 views

Show that $|I_m-AB|=|I_n-BA|$

Let $A$ be an $m\times n$ matrix and $B$ an $n\times m$ matrix. Show that $$ |I_m-AB|=|I_n-BA|. $$ I don't know where to start.
1
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1answer
25 views

Determinant of a Block Matrix times Inverse

Let $A$ be an $n\times n$ invertible matrix. Let $a$ be a number in $\mathbb{F}$, let $\alpha$ be a row $n$-tuple of numbers from $\mathbb{F}$ and let $\beta$ be a column $n$-tuple of numbers from ...
1
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0answers
20 views

Stereographic projection to show $S^n$ is a submanifold of $\Bbb R^{n+1}$

So $S^n$ in $\Bbb R^{n+1}$ can be described by the equation $x_1^2+\ldots+x_{n+1}^2=1$. Now consider two subsets $U_N:=S^n-\{(0,0,\ldots,1)\}$ and $U_S:=S^n-\{(0,0,\ldots,-1)\}$, the sphere less it's ...
1
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0answers
31 views

Linear systems, eigenvectors

For each of the following linear systems of differential equations, (i) find the general real solution (ii) show that the solutions are linearly independent (iii) draw the phase portrait a. $$\dot ...
2
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1answer
74 views

A hard exercise on endomorphisms and determinants

The following exercise has been bugging me for some days, could someone help me with it ? Let $E$ be a $\mathbb{C}$-vector space with dimension $n$ and $f\in\mathcal{L}(E)$ ($\mathcal{L}(E)$ denotes ...
0
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1answer
32 views

Prove that the determinant of this matrix is non-zero.

Prove that the determinant of this matrix is non-zero for every possible combination of + and - .$$\left[\begin{array}{cc} \pm 1 & \pm 3 & \pm 4 \\ \pm 3 & \pm 2 & \pm 5 \\ \pm 4 ...
5
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0answers
118 views

How to prove the determinant?

We have to prove the following result without expanding $\left|\begin{array}{lll} a^3 & a^2 &1 \\ b^3 & b^2 &1\\ c^3 & c^2 &1 \end{array} ...
0
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0answers
18 views

Determining matrix in terms of determinants of other matrices.

Determine |a+b e-f| |c+d g-h| in terms of the determinants of |a c| |b d| |a c| |b d| |e g| |e g| |h f| |h f| ...
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0answers
35 views

Prove the following determinant without expanding

We have to prove the following result without expanding $\left|\begin{array}{lll} a^3 & a^2 &1 \\ b^3 & b^2 &1\\ c^3 & c^2 &1 \end{array} ...
1
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0answers
53 views

Jacobian determinant of unitary transformation

Is the Jacobian determinant of a unitary transformation equal to one? I ask because I get that impression from the appendix of this paper. They have spherical coordinates for two particles, ...
0
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1answer
15 views

What is a cartesian equation for 3 space passing through 3 points?

What does cartesian equation for 3 pace look like? and is there any way to describe this equation using determinant?
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0answers
21 views

About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms)

my question is originated from a physical problem. I will try to present the problem as simple as possible, but I fear it will still be long since I'm bad at expressing myself briefly. It starts with ...
5
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4answers
248 views

Question about determinants

I am working on some practice problems and I'm unsure where to begin this problem. It starts off by giving $\det(X)= 1$ for the following matrix $X$:$$ \begin{matrix} a & 1 & d \\ b & 1 ...
5
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1answer
90 views

Most elementary proof that a determinant is divisible by $m$

So a challenge problem states that you have an $n \times n$ matrix, where each entry is an integer between $0$ and $9$, and when each row is read as a base-10 number the number is divisible by a ...
3
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1answer
76 views

Determinant of specific infinite matrix

What is the limit, as n approaches infinity, of the determinant of an n x n matrix where each cell has the value $\cos(n * row + column)$? My friend and I believe the answer to be 0, but can't ...
0
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1answer
65 views

9 by 9 matrix: finding the determinant?

Can it be done analytically? I have a system I need to solve, but would need to take a determinant of a 9 by 9 matrix. Is it worth the effort, or is there a limit (in rank) above which it's not ...
7
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0answers
189 views

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
0
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1answer
53 views

Expressing determinant as a linear combination of minors of fixed dimension

Suppose $k<n$. How does one express $\det\begin{pmatrix}a_1^1&\dots&a_n^1\\ \vdots&\ddots&\vdots\\ a^n_1&\dots&a^n_n\end{pmatrix}$ in terms of a linear combination of ...
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2answers
116 views

Give conditions on a,b,c, and d such that A has two, one, and no eigenvalues?

I am given that matrix $$A= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ and I need to find conditions on a,b,c, and d such that A has Two distinct ...
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2answers
33 views

Linear Algebra - Invertible matrices and determinants

Let $A$ be any $n \times n$ invertible matrix, defined over the integer numbers. Let assume that $A^{-1}$ (Inverse of A) is also defined over the integer numbers. Prove that $\det A\in\{-1,+1\}$. ...
1
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4answers
63 views

Let $A$ be a $3\times3$ matrix. Given $\mathrm{adj}(A)$, find $\det(A)$.

Let $A$ be a $3\times3$ matrix such that $$\mathrm{adj}(A) = \begin{pmatrix}3 & -12 & -1 \\ 0 & 3 & 0 \\ -3 & -12 & 2\end{pmatrix}.$$Find the value of $\det(A)$. I know that ...