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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5
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3answers
30 views

Calculation of determinant

Is there any easier way to make sure the determinant of the following matrix is n (the dimension of square matrix)? $ \begin{vmatrix} 1 & -1 & -1 & -1 & \cdots & -1 \\ 1 ...
1
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0answers
24 views

Characteristic Polynomial Calculation

I have a problem in my homework in which I have to find the characteristic polynomial of the following matrix: I know the final solution is: However, my answer keeps getting wrong whenever I ...
0
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2answers
21 views

Inverse of matrix sum

I found on the Wikipedia page "Determinant" the following property: For any invertible $m \times m$ matrix $X$, $\det(X + AB) = \det(X) \det(I_m + BX^{-1}A)$. Is this true? If so, how is this ...
4
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1answer
50 views

Uniqueness of determinant

In Artin Algebra 2nd edition page 22, the author proved the uniqueness of determinant by saying that any matrix $A$ can be written in reduced row-echelon form $A'$: $A'=E_1\cdots E_kA$ where $E_i$ are ...
0
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0answers
18 views

Eigenvalues of (restrictions of) the standard representation of $S_n$

Let the permutation group on $n$ elements $S_n$ act on a set $S$ of size $k < n$ via permutations. Fix some ordering on the elements of $S$ to make this sensible. Is there any way to understand ...
0
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1answer
21 views

Finding the real irrational root of a cubic polynomial?

I just wanted to check if anyone can see a simpler way to solve this. Because I am not looking forward to using the cubic formula to solve it! $$ det(\lambda-AI) = \left| \begin{array}{ccc} \lambda + ...
0
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0answers
11 views

Proof for Determinants using Laplace and induction.

Matrix $A = (a_{ij}) \in M (n x n, Field)$, Matrix $B = ((-1)^{i+j}a_{ij})$ I need to prove that det(A)=det(B). I thought induction might be one solution, but I don't know how to apply the Laplace ...
0
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0answers
22 views

Reference for the proof of interlacing of eigenvalues of submatrices

If one has a $n \times n$ Hermitian matrix $A$ and one removes $k$ of the rows and their corresponding columns then the eigenvalues of the remnant interlace the eigenvalues of the full matrix. Can ...
-1
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1answer
32 views

Prove the equality of two determinants. [on hold]

Matrix $A = (a_{ij}) \in M (n x n, Field)$, Matrix $B = ((-1)^{i+j}a_{ij})$ Proof that $det(A)=det(B)$? Thanks in advance.
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0answers
22 views

By using the properties of determinants prove that [on hold]

By using the properties of determinants, prove that:
0
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1answer
23 views

AB = Identity matrix; matrices; determinants; proof

Let $M(n\times n, \mathbb Z)$ be the set of all $n\times n$- matrices with integer coefficients, and a matrix $A \in M$. Proof, that: There is exactly one matrix $B \in M(n\times n, \mathbb Z)$ with ...
4
votes
3answers
71 views

Determinant of the inverse matrix [duplicate]

I'm seeking for a proof of the following: Let $A$ be an invertible matrix. Then the determinant of $A^{-1}$ equals: $$\left|A^{-1}\right|=|A|^{-1} $$ I don't know where to begin the proof. Any ...
2
votes
1answer
29 views

How to prove this result using Permutations? [on hold]

Let A be the set of all $3*3$ skew symmetric matrices whose entries are either -1, 0 or 1. If there are exactly 3 zeroes, three 1's and three (-1)'s, then prove that only 8 such matrices can exist.
1
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1answer
35 views

Determinant of $\lambda I + A^TA$

What properties $\lambda I + A^TA$ have? I know that $A^T A$ is positive semi-definite, and symmetric. I want to show that the determinant of $\lambda I + A^TA$ decreases as $\lambda$ increases!
1
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0answers
32 views

Determinant over $\mathbb{C}$ of an $\mathbb{H}$-linear mapping.

Let $V = \mathbb{C}^n$ and let let $u$ be a $\mathbb{C}$-linear endomorphism of $V$. Then $u$ can also be considered as an $\mathbb{R}$-linear mapping $u_{\mathbb{R}}$. It is well known that $$\det ...
0
votes
0answers
20 views

How to prove that a matrix with specific property is invertible?

If we have a square matrix $$ M = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & ...
0
votes
2answers
29 views

Determinant by nullifying

I am supposed to calculate the value for the determinant of this matrix. I didn't know what to do, so I looked up for the sample solution, which I don't understand. $$\left|\begin{array}{ccc} 18 ...
2
votes
1answer
52 views

Find the value of the Determinant

If $a^2+b^2+c^2+ab+bc+ca \le 0\quad \forall a, b, c\in\mathbb{R}$, then find the value of the determinant $$ \begin{vmatrix} (a+b+2)^2 & a^2+b^2 & 1 \\ 1 & (b+c+2)^2 ...
1
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2answers
29 views

How many solutions exist for a matrix equation $A^2=I$?

Let $A$ be a square matrix of order three or two, and $I$ be a unit matrix. How many solutions are possible for the equation $$A^2=I$$? In case the solutions are infinite, or very large, how do I ...
2
votes
0answers
23 views

Log concavity/convexity of a determinant

I was wondering if anyone would be able to help me determine whether the following quantity is log concave or not with respect to $\alpha$? $$\left[\det(\textbf Y^\top \textbf P \textbf G \textbf ...
1
vote
1answer
41 views

Determinant and matrix power

I was wondering if there is a relation between the determinant of a matrix and the determinant of its powers. I mean I am looking for something like $$ \det (A^k) = f(\det(A), k). $$ A few check I ...
0
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0answers
19 views

Transformation into a field with the result being a multiple of the determinant

Let $K$ be a field, $n \in N$ and d: $M_{n,n}(K) \to K $ an homogeneous and skew invariant transformation where $M_{n,n}(K)$ are the matrices over the field. Show that there's a $d$ with $d = c * ...
0
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1answer
24 views

Given $\det(A)$ and $\det(B)$, is my calculation of $\det(-2B^T B A)$ correct?

Suppose $A$ and $B$ are $3 \times 3$ matrices with $\det(A) = -2$ and $\det(B) = -1$. What is the determinant of $C = -2 B^T B A$? I know that $$\det(A^T) = \det(A) \qquad \det(AB) = \det(A) ...
1
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1answer
31 views

Determinant of 3 points.

I have $P=(p_1,p_2)$ and $Q=(q_1,q_2$) two points in $\mathbb R^2$, $P\ne Q$, and $R=(r_1,r_2)$ another point. What means the following determinant? $$\Delta (P, Q, R)= \begin{vmatrix} ...
1
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1answer
30 views

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = ...
0
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0answers
28 views

Determinant of matrix n x n [duplicate]

How to calculate $det\begin{bmatrix}1 & x_1 & x_1^2 \dots x_1^{n-1} \\ 1 & x_2 & x_2^2 \dots x_2^{n-1} \\ \\ 1 & x_n & x_n^2 \dots x_n^{n-1}\end{bmatrix}$?
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1answer
22 views

Determinant of 2 transpose matrix A and B.

Can you show me why $\det(A^T B^T) = \det(A)\det(B^T) = \det(A^T)\det(B)$ ? im really having a hard time finding its properties. i dont know what to search. please help.
2
votes
2answers
29 views

How to prove this result?

Let {$\Delta_1,\Delta_2,\Delta_3\cdots\cdots\cdots\cdots\Delta_n$} be the set of all determinants of order 3 that can be made with the distinct real numbers from set $S=\{1,2,3,4,5,6,7,8,9\}$. Then ...
0
votes
1answer
44 views

Correct proof of $\det(X+iY) \det(X-iY)>0$?

Can someone please look over my proof below as to why $\det(X+iY) \det(X-iY)>0$ for real matrices $X,Y$, such that $\det(X+iY)$, $ \det(X-iY)$ not both the zero, and tell me if it's correct ? My ...
4
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1answer
31 views

linearly independent and determinant

This question says a matrix $\begin{bmatrix}a & b\\c & d\end{bmatrix}$ where $a_{ij}$ are real numbers. I need to prove that $\det|A|=ad-bc\neq0 \iff $the columns are linearly independent. ...
2
votes
2answers
64 views

Block Matrix Determinant Proof

I am trying to solve the determinant of a Block matrix $$\begin{bmatrix}A-Ia&B\\B &A-Ib \end{bmatrix}$$ where a and b are integers and I is an identity matrix, A and B are square. ...
0
votes
1answer
12 views

Express the vector $b=2i-3j+5j$ in terms of these set of three vectors

The three vectors are: $$a_1=i+j+k$$ $$a_2=i-j$$ $$a_3=i+j-2k$$ I have been asked to express the vector $b=2i-3j+5j$ in terms of the three vectors above like: $b= \alpha a_1+\beta a_2+\gamma a_3$. ...
0
votes
1answer
27 views

What method is used to find the determinant of this $4 \times 4$ matrix?

This is a pre-solved example in my book, I don't understand how they solved it. What method is used? Find the determinant of $A = \begin{bmatrix} 0 & 1 & 0 & 2\\[0.3em] -1 ...
2
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0answers
47 views

Determinant of a sum

We have that: $\textbf Y \in \mathbb{R}^{n \times q}, \textbf G \in \mathbb{R}^{n \times n}, \textbf P \in \mathbb{R}^{n \times n}, \textbf Q \in \mathbb{R}^{q \times q}$. Furthermore, $\textbf G$ is ...
3
votes
1answer
30 views

Find the value of the expression-

Consider a matrix $A=\begin{bmatrix}3 & 1\\-6 & -2\end{bmatrix}$, then $(I+A)^{99}$ equals ? So how can I expand this ? The solution paper gives the answer as $I+(2^{99}-1)A$
2
votes
1answer
45 views

How to prove these two statements?

Let A,B,C,D be real matrices (not necessarily square) such that $$A^T=BCD$$$$B^T=CDA$$$$C^T=DAB$$$$D^T=ABC$$ For the matrix S=ABCD, prove that $$S^3=S$$ and $$S^2=S^4$$ My little brother got this in ...
2
votes
1answer
37 views

Maximum value of $f(x) = \log_{(\tan x + \cot x)}(\det A)$ for a diagonal matrix $A$

If $$A =\begin{pmatrix} d_1 & 0 & 0 & 0 \\ 0 & d_2 & 0 & 0\\ 0 & 0 & d_3 & 0\\ 0 & 0 & 0 & d_4\\ \end{pmatrix}$$ ...
1
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1answer
29 views

$a_{ij}=i $ if $i+j=n+1$ and $0$ otherwise; compute det $A$

The entries of the matrix is specified by this rule, $A=(a_{ij})\in M_n(\mathbb R)$, $a_{ij}=i$ if $i+j=n+1$ and $0$ otherwise. Compute det $A$ > I have seen ...
2
votes
2answers
40 views

Easiest way to calculate determinant 5x5 witx x

I would like to calculate this determinant: ...
1
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1answer
48 views

Which of the following cannot be the value of $g(x)$

Let A = $\begin{bmatrix}1 & \tan x\\-\tan x & 1\end{bmatrix}$ then let us define a function $f(x)=\begin{vmatrix}A^{T}A^{-1}\end{vmatrix}$ then which of the following cannot be the value of ...
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1answer
37 views

The value of $\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}$ for given determinant is

If $a\neq p$, $b\neq q$, $c\neq r$ and $\left|\begin{array}{cc}p&b&c\\a&q&c\\a&b&r \end{array}\right|= 0$ then the value of $\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}$ is (a) ...
7
votes
1answer
88 views

Is there a deeper meaning behind the “determinant” formula for the cross product?

We all know that for all vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R^3}$, if $(a_x,a_y,a_z)^\top$ is the component form of $\mathbf{a}$ and similarly $(b_x, b_y, b_z)^\top$ is the component form of ...
0
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3answers
94 views

Determinant of this matrix

So I have a problem.. I already found out what kind of matrix it is.. So all main diagonals of this matrix are 0.. the rest is 1.. it's not 4x4 or 3x3 etc.. it's size is nxn.. does anyone of you know ...
0
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1answer
88 views

Calculate the determinants of the following matrices

so I got this task from my professor and wanted to ask for help I have this following matrices (a) $$A = \begin{pmatrix} -3 & -11 & -11 & 45 \\ 1 & 11 & 10 & -83 \\ 1 ...
4
votes
2answers
76 views

Finding determinant of $n \times n$ matrix

I need to find a determinant of the matrix: $$ A = \begin{pmatrix} 1 & 2 & 3 & \cdot & \cdot & \cdot & n \\ x & 1 & 2 & 3 & \cdot & \cdot & n-1 \\ x ...
2
votes
0answers
23 views

Determining linear independence of three simple functions for a third order ODE. (2.9-7)

This is a very similar post to one previous by me but I felt that not all questions were satisfactorily answered. But I am sincerely grateful to those who tried. I would like a sharp independent eye ...
2
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2answers
30 views

Determining linear independence of three simple functions for a third order ODE. (2.9-6)

I would like a sharp independent eye other than my own to review my work here. I have a few questions I would like answered. Did I actually answer/solve all parts of this problem? Determinants of ...
0
votes
1answer
23 views

Determinants to solve a system

I was reading a book on Calculus when I came across this: $$\begin{cases} v+\ln(u)=xy \\ u+\ln(v)=x-y \\ \end{cases}$$ $$\begin{cases} \frac1u\frac{\partial u}{\partial x} +\frac{\partial ...
4
votes
1answer
35 views

determinant of divisor functions

Let A be a $(n-1) \times (n-1)$ matrix whose entries $a_{ij}=d(\gcd(i+1,j+1))$. $d(n)$is the number of divisors of $n$. It seems that the determinant of it is the number of square-frees less than or ...
5
votes
3answers
101 views

Find the determinant of the following matrix

Find the determinant of the following matrix: $$A = \begin{bmatrix} 1+x_1^2 &x_1x_2 & ... & x_1x_n \\ x_2x_1&1+x_2^2 &... & x_2x_n\\ ...& ... & ... &... \\ ...