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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9
votes
1answer
74 views

Prove the n-th power of a matrix is the null matrix

Let $A,B$ squared matrixes with complex elements, $dim(A)=dim(B)=n, AB=BA, \det(B)\ne0$, having the following property: $|\det(A+zB)|=1, \forall z \in \mathbb{C}, |z|=1$. Prove ...
-2
votes
3answers
54 views

Evaluate a $3\times3$ determinant. [on hold]

Show that $$\left|\begin{matrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{matrix}\right|=(a-b)(b-c)(c-a)$$
3
votes
1answer
18 views

Determinant of hankel matrix of hyperbolic functions, $a_n=\frac{n}{\sinh(\pi n)}$

I am trying to learn about the properties of Hankel matrices, and they appear to have nice closed forms for quite a large class of sequences. The class I am interested is when the elements $a_n$ are ...
1
vote
1answer
41 views

If the determinant of a matrix goes to infinity, does it means it has no inverse?

Context I have a linear time-invariant (single-input, single-output) system in state space representation (https://en.wikipedia.org/wiki/State-space_representation#Linear_systems): $$ \mathbf{x'}(t) ...
0
votes
1answer
19 views

Rule of thumb on number of zero entries for invertibility of a $4\times 4 $ matrix?

I have to determine whether a $4\times 4$ matrix $A$ is invertible. Suppose that there are no zero columns or zero rows. Is there any rule of thumb saying how many zero entries can be at most in $A$, ...
0
votes
1answer
54 views

How prove this determinant can't zero

Let $x,y,z\neq 0$ be real numbers, show that $$f(x,y,z)=\begin{vmatrix} \sqrt{x^2+y^2}&|x|&|y|\\ |y|&\sqrt{y^2+z^2}&|z|\\ |x|&|z|&\sqrt{x^2+z^2} \end{vmatrix}\neq 0$$ or it ...
1
vote
1answer
44 views

Differentiation Involving Determinant.

I have to compute the following differentiation : $$\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times ...
2
votes
2answers
32 views

How do I find value of a and b in this matrix question?

This is a question from a homework sheet my teacher gave. I already did alternate a. Alternate b is quite confusing! It asks to find the value for a and b. I don't really know what to do but here's ...
0
votes
1answer
47 views

Calculation of characteristic polynomial

I have to determine the characteristic polynomial of the matrix $$A = \begin{pmatrix} 0 & 0 &\cdots &0& -a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots ...
4
votes
1answer
71 views

Calculating the determinant of a matrix using its rank

Let A, B, C and D be real n×n matrices. If $$\operatorname{rank} \begin{bmatrix} \ A & B \\[0.3em] \ C & D \\[0.3em] \end{bmatrix} = n$$ then show that $$\det ...
1
vote
1answer
48 views

Adjoint of an adjoint of a matrix

Can you please help me on this question? $\DeclareMathOperator{\adj}{adj}$ $A$ is a real $n \times n$ matrix; show that: $\adj(\adj(A)) = (\det A)^{n-2}A$ I don't know which of the expressions ...
0
votes
0answers
18 views

Inverse Gramian matrix

Show that inverse Gramian matrix is Gramian matrix Here is my idea. $\Gamma ^ {-1} = \frac{A}{|\Gamma|}$, where A is transposed matrix of cofactors (not sure about the term, correct me please), ...
4
votes
1answer
32 views

Matrix consisting of cosines of differences

Consider the following matrix: $$\left[\begin{array}{cccc} \cos(x_1-y_1) & \cos(x_1-y_2) & \ldots & \cos(x_1-y_n) \\ \cos(x_2-y_1) & \cos(x_2-y_2) & \ldots & \cos(x_2-y_n) \\ ...
3
votes
3answers
39 views

Matrix with a certain pattern

Consider the following matrix: $$\left[\begin{array}{cccc} 1+x_1y_1 & 1+x_1y_2 & \ldots & 1+x_1y_n \\ 1+x_2y_1 & 1+x_2y_2 & \ldots & 1+x_2y_n \\ 1+x_3y_1 & 1+x_3y_2 & ...
1
vote
1answer
16 views

For integers $n>1$ , $k$ , does there exist matrix $A$ with integer entries and first row $(1,2,…,n)$ such that $\det A=k$?

Let $n >1$ be an integer , then is it true that for any integer $k$ , there exist a matrix $A \in M(n,\mathbb Z)$ with first row of $A$ as $(1,2,...,n)$ such that $\det A=k$ ?
0
votes
0answers
104 views

Prove that $ \det{\begin{bmatrix}A & B \\-B & A\end{bmatrix}}\geq 0$ [duplicate]

Let $A,B \in M_n(\mathbb{R})$. Prove that $\det{\begin{bmatrix}A & B \\-B & A\end{bmatrix}}\geq 0$. I know that there is a theorem which says that if $E,F,G,H \in M_n(\mathbb{F})$ and ...
7
votes
2answers
262 views

Non-negative determinant of a block matrix

Here's the problem I've been stuck on for some time now. Let $A,B \in M_n(\mathbb{R})$. Let $C= \begin{bmatrix} A & B \\ -B & A \\ \end{bmatrix} $ be a real ...
4
votes
1answer
58 views

Inverting an $n \times n$ matrix using determinant

We're asked to invert the following matrix with the help of guided questions. $$\begin{pmatrix} 1 + a_1 & 1 & \cdots & 1 \\ 1 & 1+a_2 & \ddots & \vdots \\ \vdots & \ddots ...
1
vote
1answer
89 views

Determinant of augmented matrices.

Let $A$ and $B$ be $n \times n$ real matrices. How can I show that $\det \begin{bmatrix} A & B \\[0.3em] -B & A \\[0.3em] \end{bmatrix} \geq 0 $?
1
vote
1answer
43 views

Derivatives using matrices good

$$\left|\begin{matrix} (1+x)^{a_1b_1} & (1+x)^{a_1b_2} & (1+x)^{a_1b_3} \\ (1+x)^{a_2b_1} & (1+x)^{a_2b_2} & (1+x)^{a_2b_3} \\ (1+x)^{a_3b_1} & (1+x)^{a_3b_2} & (1+x)^{a_3b_3} ...
1
vote
1answer
33 views

The property that det(A) = prod of A's eigenvalues, and tr(A) = sum of A's eigenvalues

Do these two properties fail to be true, if A's characteristic polynomial fails to split? If so, then do we usually work in a vector space with the ground field = $\mathbb{C}$, when we want to use ...
2
votes
0answers
21 views

Show that matrix is totally unimodular

I want to show that this matrix is totally unimodular: \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & ...
4
votes
2answers
49 views

When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$

Let 'A' be a $2 \times 3$ matrix where as B be a $3 \times 2$ matrix if $\det(AB) = 4$ the find value of the $\det(BA)$ My attempt: I took A = $$ \begin{bmatrix} 2 & 0 &0\\ ...
0
votes
0answers
17 views

Coefficient of bivariate polynomial as a determinant of matrix

Given $$ \begin{bmatrix} a\\ b\\ c\\ d\\ \end{bmatrix}=\begin{bmatrix} a_0t^3+a_1st^2+a_2s^2t+a_3s^3\\ a_4t^2+a_5st+a_6s^2\\ a_7t+a_8s\\ a_9\\ \end{bmatrix} $$ the following equation holds: $$ ...
1
vote
1answer
38 views

Rank of square matrix $A$ with $a_{ij}=\lambda_j^{p_i}$, where $p_i$ is an increasing sequence

Let $$ A = \begin{bmatrix} \lambda_1^{p_1} & \lambda_2^{p_1} & \cdots & \lambda_n^{p_1} \\ \lambda_1^{p_2} & \lambda_2^{p_2} & \cdots & \lambda_n^{p_2} \\ ...
1
vote
0answers
47 views

Given matrices $A,B, \det(A) = 2$ find $\det(B)$

Given matrices $A$,$B$, $\det(A) = 2$ Find $\det(B)$ $$A= \begin{pmatrix} a & 1 & b \\ 2 & 3a & 1 \\ b & 1 & 2a \\ \end{pmatrix}, ...
7
votes
1answer
623 views

Determinant of Matrix with uncomputable values.

Calculate the determinant of the matrix $$ \begin{pmatrix} 10^{10} & 10^{10^{10}} & 11^{11^{11}} & 1 & 0 \\ 2^{2^2} & 3^{3^3} & 7^{7^7} & 0 & 1 \\ 11 & ...
1
vote
2answers
27 views

Proving $\det \big(Df\big|_x\big)=0$ for a function into unit circle

Let $f:\mathbb{R}^2\to S$ where $S=\{x\in\mathbb{R}^2:\, ||x||=1\}$. Prove that $\det \big(Df\big|_x\big)=0$ for all $x$. I'm having trouble attacking this. So I need to show that there is some ...
0
votes
0answers
10 views

Proof attempt: A is an antisymmetric matrix (of even size). B is another matrix such that $b_{i,j}=a_{i,j}+c$. Prove that |A|=|B| [duplicate]

I asked this question but all the answers I got were outside of my scope of understanding, so here is as close as I got to a solution: $$\begin{bmatrix} c & a_{12}+c &...&&a_{1n}+c \\ ...
1
vote
1answer
24 views

tell Positive Definite Matrices by the sign of determinants

This is from my textbook I don't understand why it didn't mention the other situation which is $det(A_k) <0 $ for all k, and we stall have positive pivot because ...
3
votes
3answers
107 views

Prove that $\det(A^{T}A) \neq 0$

How to prove that $\det(A^{T}A) \neq 0$ if coloumns of $A$ are linearly independent, without using Cauchy-Binet formula? $A$ is real matrix.
1
vote
1answer
46 views

Finding determinant of matrix through row operations [problem help]?

I am having trouble understanding a problem that my Linear Algebra class gave. I understand that determinants can be found through row operations with the following points: 1.) Adding a multiple ...
0
votes
0answers
39 views

determinant of a matrix with increased off-diagonal elements

I have symmetric matrix which is formed by complex integer vectors as follows \begin{bmatrix} \|f_1\|^2 & af_2^Hf_1 & bf_3^Hf_1 & \dots \\ a^*f_1^Hf_2 & \|f_2\|^2 & cf_3^Hf_2 & ...
2
votes
0answers
45 views

Determinants Proof Can't Solve

Let $v_1,\cdots ,v_n$ be vectors in $\mathbb R^n$. Define $w_i$ as the vector in $\mathbb R^{n+1}$, which is just $v_i$ with a $0$ added to the beginning. (So if $v_i = (1,0,1)$, then $w_i = ...
-7
votes
1answer
115 views

Generalized determinant of order $n+2$

How to solve following determinant of order $n+2$ to get eigenvalues? $\begin{vmatrix} -\lambda & 0 & 1 & 1 & 1 & \cdots & 1 \\ \dfrac{n}{\lambda} & ...
0
votes
1answer
21 views

A family of vectors is linearly independent.

Let $K$ be a field and $E$ be a $K$-vector space of dimension $n$. Let $\phi$ be an endomorphism of $E$. Let $(\lambda_1,\cdots,\lambda_n)$ be a family of distinct scalars and $(x_1,\cdots,x_n)$ be ...
3
votes
0answers
20 views

Determinant of $\delta$ function

Let $$\delta_i^j=\left\{ \begin{aligned} 1 ~~~~~~i=j \\ 0 ~~~~~~i\ne j \end{aligned} \right. $$ $1\le i,j\le n$. How to prove $$ \begin{vmatrix} \delta_{j_1}^{i_1} ~...~ \delta_{j_n}^{i_1} \\ \\ ...
0
votes
1answer
16 views

Show that every curvature of a Frenet curve satisfy the following statement.

I need to show the following statement: Show that for every Frenet curve $c:I\to\mathbb{R}^n$, the curvatures $\kappa_1(t),\ldots,\kappa_{n-1}(t)$ satisfy the following equality: ...
0
votes
0answers
24 views

Inductive step determinant proof

I need help with this inductive step. Assume that the determinant function for an N $x$ N matrix exists and fulfills the three properties of a determinant. For $\{v_1, ... , v_n\}$ in $\mathbb{R}^n$, ...
1
vote
1answer
30 views

determinants of matrix with adjoints of order 2

Let $A$ be a square matrix of order $2$ with $\lvert A \rvert\ne 0$ such that $\big\lvert A+\lvert A \rvert \operatorname{adj} (A)\big\rvert=0$, then the value of $$\big\lvert A-\lvert A \rvert ...
4
votes
1answer
43 views

$\det(I+A)$= sum of all principal minors of $A$

I'm having a hard time proving or finding a proof for the following result. It should follow from an application of the Laplace expansion. Let $n\in\mathbb{N}$, $[n]=\{1,\dots,n\}$, and ...
0
votes
1answer
24 views

Given two column vectors $a$ and $b$, what is the determinant of $A$ if $A=Id-ab^T$

Given two column vectors $a$ and $b$ in $\mathbb R^n$ , $n \ge 2$, form the $n×n$ matrix and $I_n$ the identity matrix. Let be $A = I_n-ab^T$. What is the determinant of $A$?
0
votes
0answers
12 views

Linear independence in a module

It is widely known that for any matrix on a commutative field, the following properties are equivalent : 1. Determinant is invertible 2. Matrix has an inverse 3. The only zero linear combinations ...
2
votes
1answer
50 views

How can I solve for a , b , c , d?

Let's say I fix a list of two real numbers $\sigma = (\sigma_1, \sigma_2)$, and I want to show that there exists a real, entrywise-nonnegative matrix $A$ with $\sigma$ as its spectrum. How could I ...
0
votes
2answers
28 views

The determinant of the transposing endomorphism

Let $K$ be a field and $f$ the endomorphism of $\mathcal M_n(K)$ that sends a matrix to its transpose. I want to determine the determinant of $f$. I know that since $f^2=id$ then $det(f)=1\ or \ -1 $ ...
5
votes
2answers
78 views

If $BA = I$, prove that $AB = I$ (using determinants)

I've seen this problem around here, but I wanted to check if this particular solution is right. So, if $BA = I$, then $det(B)det(A) = 1$, meaning neither $det(B)$ or $det(A)$ are equal to $0$. ...
0
votes
2answers
53 views

With these two equations, how do I show that either a,b,c,d must be negative, if v is not 0?

If I have the equations $$ad-bc = u^2 +v^2$$ $$a+d = 2u$$ and I want $a, b, c, d \ge 0$, then how I can show that this is impossible, if $v \ne 0$? I.e., if $v \ne 0$, then one of $a,b,c,d$ must ...
6
votes
2answers
81 views

What is the determinant of []? [closed]

I typed this in Matlab, but I can't understand why it returns the determinant one. A = [] det(A) ans = 1
1
vote
3answers
53 views

Prove that the product of two invertible matrices also invertible

I am working on a homework problem, but I am lacking some understanding. Here is the problem: Let $A$ and $B$ be invertible $n \times n$ matrices with $\det(A) = 3$ and $\det(B) = 4$. I know that ...
1
vote
2answers
43 views

Determinant of map $p(x) \mapsto (Tp)(x)=a_n+a_{n-1}x+ \ldots +a_0x^n$

Let $V$ be the vector space of polynomial $\mathbb{R}$ of degree less than or equal to $n$. For $p(x)=a_0+a_1x+ \ldots +a_nx^n$ in $V$. Define a Linear Transformation $T:V \to V$ by ...