Tagged Questions

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.

learn more… | top users | synonyms

0
votes
1answer
37 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
votes
0answers
12 views

volume parallelepiped-linear algebra

So I have this exercise where they give me the vertices and i must chose those to use to calculate the volume= absolute value of the transformation matrix. The matrix will be $3\times 3$, so i am ...
0
votes
2answers
38 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
votes
2answers
57 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
vote
1answer
28 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
46 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 $$ \mathrm{det}(I_m-AB)=\mathrm{det}(I_n-BA). $$ I don't know where to start.
1
vote
1answer
23 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 ...
0
votes
0answers
9 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
vote
0answers
26 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
votes
1answer
51 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
votes
1answer
24 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 ...
4
votes
0answers
63 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
votes
0answers
15 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| ...
1
vote
0answers
30 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
vote
0answers
22 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
votes
1answer
12 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?
0
votes
1answer
37 views

How do we establish the existence of fundamental matrix of a Markov chain?

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 ...
0
votes
0answers
17 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
votes
4answers
205 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
votes
1answer
76 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 ...
1
vote
0answers
25 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 figure ...
0
votes
1answer
51 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 ...
6
votes
0answers
154 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
votes
1answer
48 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 ...
1
vote
2answers
102 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 ...
1
vote
2answers
30 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
vote
4answers
54 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 ...
1
vote
4answers
76 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.. ...
0
votes
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 ...
1
vote
0answers
28 views

Differential equations, derivative of determinant, Euler's formula

Let $b:\mathbb{R}^n\to\mathbb{R}^n$ be a smooth vector field. Let $u(s,x,t):\mathbb{R}^{n+2}\to\mathbb{R}^n$ with $s,t\in\mathbb{R}$ and $x\in\mathbb{R}^n$ satisfy the following differential ...
1
vote
4answers
91 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 ...
4
votes
1answer
153 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
votes
1answer
17 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 ...
1
vote
1answer
26 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} & ...
1
vote
3answers
41 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
votes
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 ...
10
votes
3answers
142 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
votes
1answer
55 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) ...
2
votes
1answer
19 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
votes
0answers
26 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
votes
1answer
40 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} & & & \\ ...
7
votes
2answers
117 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
votes
1answer
43 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
votes
2answers
56 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 $\det (xy^t)=0$ ?
3
votes
1answer
60 views

Find the determinant of a symmetric matrix

How can we find the determinant of the following matrix $A$: $\left( \begin{array}{cccccc} x_1y_1 & x_1y_2 & x_1y_3 & \cdots & x_1y_{n-1} & x_1y_n \\ x_1y_2 & x_2y_2 & ...
0
votes
0answers
45 views

invertible matrix and upper triangular matrix

If we are given a $A$ as $2\times2$ matrix. How to find an invertible matrix $P$ and a upper triangular matrix $U$ such that $A=PUP^{-1}$?
1
vote
0answers
142 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.
0
votes
1answer
26 views

odd determinants

How many 4 × 4 matrices with entries from {0, 1} have odd determinant? is there a short way of finding the answer to this question or do we have to solve it by hit and trial or using lengthy methods. ...
0
votes
1answer
24 views

How do i prove the Leibniz formula of the determinant over a commutative ring?

Let $R$ be a commutative ring. My definition for the determinant over $M_n(R)$ is defined inductively as $\det_{n+1}(A)=\sum_{j=1}^n (-1)^{1+j}A_{1j} \det_n(\tilde{A_{ij}})$. (Here, $(-1)$ denotes ...
0
votes
1answer
29 views

calculating determinant

Let M be real vector space of order $2\times3$ matrices with the real entries. Let $T:M\longrightarrow M$ be defined by $T\Bigg( \begin{pmatrix} x_{1} & x_{2} & x_{3} \\ x_{4} ...