0
votes
0answers
33 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
0
votes
1answer
19 views

the volume of pyramid value

when calculating the volume of pyramid using a determinnat, is it ok to take the determinanat in absloute value so that every negative result would be converted to positive volume number?
0
votes
0answers
40 views

Applying properties of determinants [closed]

I got numbers 2 and 4 and 5 6 but i cant seem to understand or get 1 and 3...i know this might be dumb and silly but i am stuck on these 2 last questions. My problem is that i dont understand what do ...
1
vote
1answer
62 views

Compute the determinant $4\times 4$

Compute the determinant: $$ A= \begin{vmatrix} 1 & 1 & a+1 & b+1 \\ 1 & 0 & a & b \\ 2 & b & a & b \\ 2 & a & a ...
1
vote
2answers
33 views

Determine the values of $k$ so that the following linear system has unique, infinite and no solutions.

Determine the values of $k$ so that the following linear system has a unique solution, infinite solutions and no solution. $2x + (k + 1)y + 2z = 3$ $2x + 3y + kz = 3$ $3x + 3y − 3z = 3$ I have ...
1
vote
4answers
56 views

Find matrix determinant

How do I reduce this matrix to row echelon form and hence find the determinant, or is there a way that I am unaware of that finds the determinant of this matrix without having to reduce it row echelon ...
2
votes
1answer
28 views

Prove that determinant of matrix equal to n

Prove that determinant of matrix $D_n$ (square $n$ x $n$ matrix) is equal to $n$. $$ \begin{matrix} 1 & -1 & -1 & \cdots & -1 \\ 1 & 1 & & & \\ 1 & & 1 & ...
5
votes
1answer
93 views

A challenge question in determinant of real matrices!

Suppose that $n\in \mathbb N -\{1\}$ and $a_{11},a_{12},\ldots,a_{nn}$ are $n^2$ distinct real numbers, prove that there is some enumeration of $a_{ij}$'s like $b_{ij}\ (i,j=1,2,\ldots,n)$ such ...
7
votes
2answers
270 views

Circulant determinants

Suppose that $a_1,a_2,\ldots,a_n$ are $n$ distinct real numbers; is the following statement true? There is a permutation of $a_1,a_2,\ldots,a_n$, namely $b_1,b_2,\ldots,b_n$, such that the ...
2
votes
1answer
112 views

A Beautiful Determinant!

Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly, $$ \begin{bmatrix} a_1 & a_2 & a_3 & \cdots & a_n\\ a_2 & a_3 & a_4 & \cdots ...
1
vote
1answer
50 views

Find Determinant of A

I've tried creating a triangular matrix, tried row reducing but can't figure it out as I keep on having c-unknown in my answer. How would I do this?
2
votes
3answers
62 views

Evaluate determinant of an $n \times n$-Matrix

I have the following task: Let $K$ be a field, $n \in \mathbb{N}$ and $a,b \in K^n$. Evaluate the determinant of the following matrix: $$\begin{pmatrix} a_1+b_1 & b_2 & b_3 & \dots ...
7
votes
1answer
124 views

Prove that determinant of matrix equal $\pm1$ or $0$

We are given square binary matrix $A_n$. Data contained by A comply the following rule: if row has any 1's then they would appear there only successively (row $(1\space 1\space0\space1 )$ is ...
0
votes
2answers
25 views

Determine all values of $k$ for which the following matrices are linearly independent in $M_{22}$

If we express these matrix vectors as an augmented matrix, we get a row of zeros. If take out this row of zeros we are left with a $3x3$ matrix, is this allowed? We can find values for which the ...
0
votes
1answer
35 views

Exploring Determinants of Matrices. [closed]

I have a homework and i have to explore different patterns of determinant. I have find a unique pattern with determinants and make a conjecture. Your ideas about different patterns will be welcomed. ...
0
votes
1answer
44 views

Proof of Nonnegative Determinant

How can I solve this question?
2
votes
1answer
33 views

Which is the max value of the determinant with 4 lines and 4 colums ,where every term is +- 1?

I understand that this problem can be solve with the volume of a tetrahedron. But i don't know how. please help me ! \begin{vmatrix} \pm1 & \pm1 & \pm1 & \pm1 \\ \pm1 & \pm1 & ...
0
votes
3answers
103 views

Proof using properties of determinants

I have to use properties of determinants to show that $$\left| \begin{array}{ccc} b^2+c^2 & ab & ac \\ ab & a^2+c^2 & bc \\ ac & bc & a^2+b^2 \end{array} \right| = ...
2
votes
1answer
88 views

Calculating the determinant gives $(a^2+b^2+c^2+d^2)^2$?

I need to calculate the following determinant in order to prove the following equality: $$\det\begin{pmatrix} a & b & c & d \\ -b & a & -d & c \\ -c & d & a & -b ...
3
votes
3answers
56 views

Vandermonde determinant for order 4

I'd like to show the case $n=4$ for the Vandermonde-determinant. It should look like this: $V_4 := \det \begin{pmatrix} 1 & 1 & 1 & 1 \\ x_1 & x_2 & x_3 & x_4 \\ x_1^2 & ...
2
votes
1answer
20 views

Proof x \in L \leftrightarrow det(…) = 0.

I just need some help with the following proof: Let $v = (v_1,v_2) $and $ w=(w_1,w_2)$ be two points in $K^2 , v \not= w$ and $L \subseteq K^2 $ a line through these two points. Show that ...
0
votes
2answers
56 views

Is this map an isomorphism?

Let $f : M_{2 \times 2} \to \Bbb{R}$ be given by $$ \{ \{ a, b \}, \{ c, d \} \} \mapsto ad-bc $$ To prove something is an isomorphism it has to be 1-1, onto and preserve structure. Can someone ...
2
votes
2answers
64 views

Calculate determinant [closed]

I have tried to do this one two times, failed both. Correct answer is $$-90.$$ Here are my attempts. The matrix in question is $$ \left[ \begin{array}{c} 1 & 3 & -1 & 0 & 2 \\ 0 ...
0
votes
0answers
47 views

Given the matrix, find c such that det(D)=0 has a repeated solution

Given the matrix $D= \begin{vmatrix} 1 & x & x^2\\ 2 & c & 4\\ 3 & 2 & 1 \end{vmatrix} $ Find c such that $\det(D)=0$ has a repeated solution for $x\in R$. I got up to ...
3
votes
1answer
59 views

Find signature of symmetric block matrix, given the diagonal blocks are positive / negative definite - Check my proof

This may be a basic question, but I'd like someone to double check it. We are given the matrix $A=\begin{pmatrix} A_1 & C \\ C^T & A_2\end{pmatrix}$ where $A_1$ is a $k$ by $k$ positive ...
0
votes
1answer
30 views

Difficulty proving formula containing the adjugate and determinant of a matrix

This is what I need to prove: You have an invertible matrix $A \in M_3(\Bbb R^3)$. Prove that $\operatorname{adj}(\operatorname{adj}(A))=\det{(A)}^{n-2}A$ The proof goes as follows: ...
3
votes
1answer
62 views

Prove $\frac{c_n(a_1,…,a_n)}{c_{n-1}(a_2,…,a_n)}=a_1 + \cfrac{1}{a_2 + \cfrac{1}{\ddots + \cfrac{1}{a_{n-1}+\frac{1}{a_n}}}}$

For $n>0$ and $a_1,...,a_n \in K$ let $c_n(a_1,...,a_n)$ be the determinant of the matrix $$ \begin{pmatrix} a_1 & 1 & 0 & \cdots & 0 \\ -1 & a_2 & \ddots & ...
4
votes
2answers
171 views

Determinant of a matrix with generalized binomial coefficients

Let $$ A= \begin{bmatrix}\binom{-1/2}{1}&\binom{-1/2}{0}&0&0&...&0\\ \binom{-1/2}{2}&\binom{-1/2}{1}&\binom{-1/2}{0}&0&&...\\...&&&\binom{-1/2}{0}\\ ...
4
votes
2answers
86 views

$T=-T^{*}$, show that $T+\alpha I$ is invertible.

Please don't answer the question. Just tell me if I am in the right direction. I should be able to solve this. We are given $T=-T^{*}$, show that $T+\alpha I$ is invertibe for all real alphas that ...
0
votes
2answers
80 views

Calculate determinant of matrix

Calculate the determinant of this matrix for $a, a_0,...,a_{n-1} \in K$ over any field $K$ $$ \begin{pmatrix} a & 0 & \cdots & 0 & -a_0 \\ -1 & \ddots & \ddots & \vdots ...
3
votes
2answers
52 views

Special Gram's inequality

For $1 \le s < k$ and $v_1$, $v_2,\dots,v_k$ vectors in $\mathbb{R}^n$, show that $$\det G(v_1, v_2,\dots,v_k) \le \det G(v_1,v_2,\dots,v_s)\det G(v_{s+1}, v_{s+2},\dots,v_k).$$ Here, $G(v_1, ...
2
votes
2answers
430 views

Finding determinant using properties of determinant without expanding [duplicate]

show that determinant $$\left|\matrix{ x^2+L & xy & xz \\ xy & y^2+L & yz \\ xz & yz & z^2+L \\ }\right| = L^2(x^2+y^2+z^2+L)$$ without expanding by ...
1
vote
3answers
138 views

If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?

If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue? I don't really know where to start with this one. I know that ...
3
votes
0answers
54 views

Proof: Determinant of a block matrix [duplicate]

My homework is due tomorrow (12h left), that means I've already lost, but I'm looking genuinely for a possible solution. The Problem: Let $n \in \mathbb{N}$ and $1 \leq r \leq n$. Let $A = ...
0
votes
0answers
266 views

Matrix with trig functions and Cramer's rule

Using Cramer's rule solve for $x'$ and $y'$ in term of $x$ and $y$ $x = x'\cos\theta - y'\sin\theta\\ y = x'\sin\theta + y'\cos\theta$ So what I have is this $\det\begin{bmatrix} \cos\theta& ...
3
votes
3answers
297 views

Determinant from matrix entirely composed of variables

I don't want the answer, but I'd love to kick in the right direction. I'm really not sure how to approach this question. $$\begin{align} & -6 = det\begin{bmatrix} a & b & c \\ d & e ...
1
vote
1answer
52 views

$n\times n$ matrix determinant of rook configuration

We place $n$ rooks on an $n\times n$ chessboard in such a way that they don't threaten each other. To each such placement corresponds an $n\times n$ matrix in which there is a $1$ at the position of ...
1
vote
1answer
264 views

Prove the following determinant identities without expanding the determinants

a) $$\begin{vmatrix} \sin^2 x & \cos^2 x & \cos 2x \\ \sin^2 y & \cos^2 y & \cos 2y \\ \sin^2 z & \cos^2 z & \cos 2z \\ \end{vmatrix} = 0;$$ $$\begin{vmatrix} \sin^2 x ...
0
votes
1answer
41 views

Compute the determinant

a) ...
0
votes
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 ...
1
vote
2answers
511 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
vote
2answers
71 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 ...
6
votes
4answers
444 views

$\det(I+A) = 1 + tr(A) + \det(A)$ for $n=2$ and for $n>2$?

Let $I$ the identity matrix and $A$ another general square matrix. In the case $n=2$ one can easily verifies that \begin{equation} \det(I+A) = 1 + tr(A) + \det(A) \end{equation} or \begin{equation} ...
0
votes
1answer
57 views

I solved the question. But I am asking a little bit. $\det(D(fog)(a))=?$

After here, how can I show its determinant?
5
votes
3answers
8k views

Determinant of symmetric matrix

Given the following matrix, is there a way to compute the determinant other than using laplace till there're $3\times3$ determinants? \begin{pmatrix} 2 & 1 &1 &1&1 \\ 1 & 2 ...
3
votes
2answers
130 views

Calculating determinant of matrix

I have to calculate the determinant of the following matrix: \begin{pmatrix} a&b&c&d\\b&-a&d&-c\\c&-d&-a&b\\d&c&-b&-a \end{pmatrix} Using following ...
2
votes
2answers
256 views

Gram determinant

How to prove that $$\sqrt{\Gamma(\vec{a},\vec{b},\vec{c})}=|(\vec{a},\vec{b},\vec{c})|,$$ where $\Gamma(\vec{a},\vec{b},\vec{c})= \left | \begin{array} {ccc} \vec{a} \cdot \vec{a} & \vec{b} ...
3
votes
1answer
110 views

Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$

Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$ Using $\det(Q)\det(\bar{Q}^T) = I$, I get to the stage $\det(\bar{Q})\det(Q)=1$, but can't do much else with it. Thanks for ...
1
vote
0answers
295 views

Determinant of a matrix with variables in it

Assuming that $z \neq 0$, compute the determinant $d_n(z) = \det D_n \left(1, z, 1 - \frac{1}{z^2} \right)$, where $z$ is a complex variable. In particular, compute the value $d_n(\sqrt{2})$. ...
1
vote
1answer
155 views

prove that determinant is a quadratic form

let $V$ be a vector space of all $2 \times 2$ hermitian matrices with entries from $\mathbb C$, over the field $\mathbb R$. prove that $q(v)=\det(v)$ is a quadratic form. I tried to prove that ...