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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1answer
17 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!
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11 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 ...
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11 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 & ...
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2answers
25 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
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1answer
49 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 ...
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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 ...
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0answers
22 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 ...
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1answer
36 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 ...
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18 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 * ...
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1answer
22 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) ...
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1answer
27 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} ...
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1answer
28 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} = ...
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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.
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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 ...
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1answer
40 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
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2answers
57 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. ...
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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: $b= \alpha a_1+\beta a_2+\gamma a_3$. ...
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1answer
25 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 ...
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44 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 ...
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16 views

Determinant of a matrix equality [closed]

What is the reason that the det(A)= if we pick random row or column and we start finding the adjugate amounts multiplied by an element and sum them?
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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 $I+(2^{99}-1)A$
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1answer
44 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 ...
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1answer
35 views

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

If $$A =\begin{pmatrix} d_1 & 0 & 0 & 0 \\ 0 & d_2 & 0 & 0\\ 0 & 0 & d_3 & 0\\ 0 & 0 & 0 & d_4\\ \end{pmatrix}$$ ...
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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 ...
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2answers
40 views

Easiest way to calculate determinant 5x5 witx x

I would like to calculate this determinant: ...
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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) ...
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1answer
84 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 ...
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3answers
93 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 ...
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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 ...
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2answers
74 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 ...
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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 ...
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2answers
29 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 ...
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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
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1answer
34 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 ...
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3answers
100 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\\ ...& ... & ... &... \\ ...
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1answer
19 views

Understanding Overdetermined System

Consider a system of linear equations $$A \times x = B$$ The system has a unique solution exactly when the determinant of the coefficient matrix (i.e. A) is nonzero. When the determinant of the ...
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Vandermonde determinants quotient [duplicate]

I have to prove that for any integers $k_1<k_2<...<k_n$ the quotient: $$ \frac{V_n (k_1,k_2, ..., k_n)}{V_n (1, 2, ..., n)} $$ is an integer, where: $$ V_n (k_1,k_2, ..., k_n) = \prod_{1 ...
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1answer
35 views

Determinant of Hermitian Tridiagonal Matrix with Constant Upper and Lower Diagonals

I got this equation where the a terms are known but I want to determine a relationship between the b terms (so, no numerical analysis please). I know that the bi terms are real and the a terms are ...
4
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2answers
221 views

Proving two determinants are equal

Using determinant properties (without expanding), prove that $$ \begin{vmatrix}yz & z^2 & y^2 \\ z^2 & xz & x^2 \\ y^2 & x^2 & xy \end{vmatrix} = xyz\begin{vmatrix}x & z ...
2
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1answer
25 views

Closed form of even indexed Euler Numbers

Prove that: $${\rm{E}}_{2n}=(-1)^n(2n)!\left|{\begin{array}{ccccccc} \frac{1}{2!} & 1 & 0 & 0 & \cdots & 0 & 0\\ \\ \frac{1}{4!} & \frac{1}{2!} & 1 & 0 & \cdots ...
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36 views

Proving generalized Cassini's identity using determinant?

Motivation It is not hard to show, by using the general solution, that Proposition. If $(a_{n})_{n\in\Bbb{Z}}$ satisfies the recursive formula $ a_{n+2} = pa_{n+1} + qa_{n}$, then for any $n, i, ...
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1answer
31 views

Determinant of sum of squares of commuting matrices

I have the following question from a math competition, can anyone help me solve this: Let $A,B\in M_n(\mathbb{R})$ be two commuting matrices ($AB=BA$). Prove that $\det(A^2+B^2)\ge0$. Thanks in ...
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2answers
65 views

Please explain definition of determinant using permutations?

Many people (in different texts) use the following famous definition of the determinant of a matrix $A$: \begin{align*} \det(A) = \sum_{\tau \in ...
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46 views

How to interpret some matrix lemmas on Wikipedia - the number 1 vs. the matrix I

I'm reading some lemmas on Wikipedia, eg, the Matrix determinant lemma, and the Sherman-Morrison formula, and both of these formulas have a 1 added to a product of column vectors and matrices. How ...
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1answer
28 views

Given a matrix $A$ of rank $n$, show that $\det(\operatorname{adj}(A))=\det(A)^{n-1}$ and $\operatorname{adj}(\operatorname{adj}(A))=(\det A)^{n-2}A$ [duplicate]

$\newcommand{\adj}{\operatorname{adj}}\newcommand{\rank}{\operatorname{rank}}$If $\adj(A)$ denotes the classical adjoint and we are given that, for an $n \times n$ matrix $A$ over $\mathbb{R}$, ...
0
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1answer
30 views

Geometric interpretation of determinant of a system of homogenous linear equations

What is the geometric interpretation of the determinant of a matrix representing a system of homogenous linear equations? We know that iff the determinant is equal to zero the system has a ...
3
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2answers
58 views

Calculating the determinant of $A$ with $A_{ij}=a$ for $i<j$, $A_{ij}=-a$ for $i>j$, $A_{ii}=x$, using a pen and paper

Let $$A = \left[\begin{array}{cccccc} x&a&a&a&\dotsm&a\\ -a&x&a&a&\dotsm&a\\ -a&-a&x&a&\dotsm&a\\ -a&-a&-a&x&\dotsm&a\\ ...