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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36 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.
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1answer
36 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 ...
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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 ...
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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
27 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
36 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, ...
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0answers
22 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
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2answers
46 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 ...
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1answer
44 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 $-1000$. ...
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2answers
45 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?
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2answers
64 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 ...
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1answer
33 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
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2answers
77 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.
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1answer
24 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 ...
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0answers
16 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 ...
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0answers
30 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 ...
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1answer
70 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 ...
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1answer
29 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 ...
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0answers
116 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} ...
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0answers
17 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
33 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} ...
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0answers
44 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, ...
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1answer
13 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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65 views

How do we establish the existence of fundamental matrix of a Markov chain? [on hold]

Let $X=\{X_n\}$ be a finite state Markov Chain with the state space $S = \{0,1,2,...,N\}$ such that $0$ is the single absorbing state and all the rest states are transient. The following is the ...
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19 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 ...
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4answers
239 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
88 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
75 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
60 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 ...
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0answers
185 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 ...
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1answer
50 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
114 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 $\det A\in\{-1,+1\}$. ...
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4answers
62 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 ...
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4answers
83 views

Use row reduction to show that the determinant is equal to this variable.

Show determinant of: \begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix} is equal to $(b - a)(c - a)(c - b)$ I'm not sure if you can use squares or square roots hmmm.. ...
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2answers
35 views

Show that a determinant is equal to this variable.

Show that the : determinant of: \begin{pmatrix}0&0&a_{13}\\0&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix} is equal to $-A_{13}A_{22}A_{31}$ I believe the cofactor and ...
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4answers
94 views

Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$

Suppose that $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging the first two rows of $A$. Prove that $\det(B) = -\det(A)$. By general $n \times n$ matrix, I mean ...
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1answer
165 views

Proof of the conjecture that the kernel is of dimension 2

I already asked this question which has been answered. This question may seem very similar but the required matrix manipulations are probably very different here due to the addition of the matrix ...
0
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2answers
24 views

Coordinate dependence of the volume of parallelotope

It is well known that for $n$ vectors $v_1, \ldots, v_n$ in $\mathbb R^n$, the determinant of the matrix $A = (v_1 \ldots v_n)$ [i.e. with the vectors as columns] is related to the volume of the ...
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1answer
27 views

Negative determinant

Let $$ A = \begin{bmatrix} -a_{12}-a_{13}-a_{14} & a_{12} & a_{13} & 1\\ a_{21} & -a_{21}-a_{23}-a_{24} & a_{23} & 1\\ a_{31} & a_{32} & -a_{31} - a_{32} - a_{34} & ...
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3answers
44 views

Determinant of linear transformation

Given a linear transformation $T:V\rightarrow V$ on a finite-dimensional vector space $V$, we define its determinant as $\det([T]_{\mathcal{B}})$, where $[T]_{\mathcal{B}}$ is the (square) matrix ...
3
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1answer
63 views

Determinant of the matrix $\binom{m_i}{j-1}$

Let $m_1,\dots,m_n$ be real numbers $\ge n-1$. How can I find the determinant of the matrix $A$ defined by $(a_{i,j})=\binom{m_i}{j-1}$, for $1\le i\le n$ and $1 \le j \le n$ ? This all looks ...
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3answers
151 views

Determinant of $a_{i,j}=(x_i+y_j)^k$

How can I find the determinant of the matrix $A\in\mathcal{M}_n(\mathbb{R})$ with coefficients $a_{i,j}=(x_i+y_j)^k,k<n$ ? All the $x_u,y_u$ are real numbers. Derivating won't help, and I didn't ...
5
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1answer
58 views

Determinant of a matrice $a_{ij}=e^{a_ib_j}$

1) Let $a_1<\dots<a_n$ real numbers and $\lambda_1,\dots,\lambda_n\in\mathbb{R}\backslash\{0\}$ Let $f(x)=\lambda_1e^{a_1x}+\dots+\lambda_ne^{a_nx}$ Show that $f$ has at most $n-1$ zeroes 2) ...
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1answer
20 views

Determinant of block matrix when $CD^T=DC^T$

When $CD^T=DC^T$ and $D$ is invertible we have: $$\left(\begin{array}{cc} A & B\\ C & D\\\end{array}\right)\times\left(\begin{array}{cc} D^T & 0\\ -C^T & ...
0
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0answers
28 views

Matrix Inverse Question- Singular Matrix issue

I have a given Matrix equation $R(s)^{'}_{3\times 3} = \psi(s)_{3\times 3}R(s)\tag 1$ Conditions R(s) is orthogonal and determinent 1. Can say in the format of rotation matrix $R^{'}(s)$ ...
0
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1answer
42 views

Do Tensors have a determinant property?

We know that only square $n \times n$ matrices have a determinant property! And it can be defined just like this: $$A=\begin{array} & & & \\ ...
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2answers
125 views

Sum of squares of maximal minors of a rectangular matrix with orthonormal rows

A matrix $A$ has $m$ rows and $n$ columns, such that $m \leq n$. We know that each row of $A$ has norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is ...
0
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1answer
48 views

Demonstrate using determinant properties that the determinant of matrix “A” is equal to, 2abc(a+b+c)^3

How can I show, using determinant properties of matrix, that: \begin{equation} \det\begin{pmatrix}(b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & ...
0
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2answers
57 views

To prove $\det (xy^t)=0$ [duplicate]

Let $x,y$ be arbitrary non-zero column vectors in $\mathbb R^n$ , then how do we prove that $\det (xy^t)=0$ ?