Questions tagged [determinant]

Questions about determinants: their computation or their 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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13 views

Definition of second cofactor of a matrix

I am reading a paper and I came across with the term a second cofactor of matrix $M$ Let $C_{ab}$ be the operator cofactor, that is $C_{ab}$(M) is the cofactor o matrix $M$ . Is this the definition ...
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11 views

$ \frac{1}{24xyz} ( {A}_{1}- {A}_{2}) $ is equal to

Suppose $xyz \ne 0$ and $ {A}_{1} = $ \begin{vmatrix} (1+2x)^{2} & (1+2y)^{2} & (1+2z)^{2} \\ (1+3x)^{2} & (1+3y)^{2} & (1+3z)^{2} \\ (1+4x)^{2} & (1+4y)^{2} & (1+4z)^{2} \...
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1answer
18 views

Linear independence of vectors and minor of matrices

Prove that if a minor of order $k$ is nonzero, then the corresponding columns of the matrix are linearly independent "The rank of a matrix is the maximal order of a nonzero minor of $A$" ...
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31 views

How can I find the determinant of a $n\times n$ square matrix whose all diagonal elements are equal to $a$ and off-diagonal elements are equal to $b$? [duplicate]

How can I find the determinant of this type of square matrix of $n\times n$ $$\begin{vmatrix} a&b&b&b& \ldots&b\\b&a&b&b&\ldots&b\\b&b&a&b&\...
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1answer
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Not able to factorize determinant

I know how to solve this by determinant properties but I can't find any intuitive way to solve this by factorization method. $$\Delta=\left|\begin{matrix}b^2+c^2&ab&ac\\ab&c^2+a^2&bc\\...
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2answers
106 views

Calculate the determinant of $n^\text{th}$ order

Calculate the determinant of $n^\text{th}$ order: $$ \begin{vmatrix} 1 + a_1 & 1 + a_1^2 & \dots & 1 + a_1^n \\ 1 + a_2 & 1 + a_2^2 & \dots & 1 + a_2^n \\ \vdots & \...
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The measure of the image of a linear map

A somewhat similar question was already asked but I'm trying to prove it a different way. I'm somewhat new to the website so not sure if I should've commented on the old thread or not but here it is. ...
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1answer
28 views

How to evaluate the maximum and minimum values of a determinant? [closed]

The determinant is given by $\begin{vmatrix} k-i & l-j \\ k-m & l-n \end{vmatrix}$ with $(i,j) \neq (k,l) \neq (m,n)$; $0 \leq i,k,m \leq d_1$ and $0 \leq j, l, n \leq d_2$
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1answer
81 views

Show $\log(\det(A))\le \operatorname{tr}(A)-n$

Suppose that $A$ is a real, symmetric, positive definite $n\times n$ matrix. Show that $$\log(\det(A))\le \operatorname{tr}(A)-n \quad \text{and} \quad \log(\det(I_n+A))\le \operatorname{tr}(A).$$ ...
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How to prove the determinate of the cartan matrix of a group ring is positive?

I'm reading the text Representations of Finite Groups by Nagao and Tsushima. In an exercise, it asked to prove $\det C > 0,$ where $C$ is the Cartan matrix of a group ring $RG$ and $R$ is a ...
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42 views

How to efficiently calculate Vandermonde's determinant?

The determinant of Vandermonde matrix $$V=\left[\begin{matrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\ 1 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\ 1 & x_3 & x_3^2 &...
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1answer
32 views

How to prove this equation about calculation of matrix determinant?

How to prove the equation about the determinant of Matrix $M$, i.e., $|M|=\frac{(M \cdot a) \times (M \cdot b) \cdot (M \cdot c)}{a \times b \cdot c}$ where $a$, $b$ and $c$ are arbitrary vectors. ...
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Equality of ideals generated by matrix minors

Let $A,B\in M_n(R)$ matrices over commutative ring. We say that $A\sim B \Leftrightarrow B=PAQ$ where $P,Q$ are invertible. Denote by $\Delta_k(A)$ the ideal in $R$ which is generated by all the ...
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Show that $\det(A_n) =(a-1)^{n-1}(a+n-1)$ [duplicate]

Show that $\det(A_n) =(a-1)^{n-1}(a+n-1)$ with K as a field and $ n \in N$ $ A_n = \begin{pmatrix} a & 1 & 1& 1 & ... & 1 \\ 1 & a & 1 & 1 & ... & 1 \\ 1 & ...
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73 views

How to prove $\det(𝐼−𝐴𝐵)=\det(𝐼−𝐵𝐴)$? [closed]

prove that if $A$ and $B$ are $n\times n$ matrices then $\det(I-AB)=\det(I-BA)$
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Determinant formula manipulation

Let $P\in M_{k\times n}$ and $A\in M_{n\times k}$ then $$det(PA)=\sum_{\sigma\in S_k}\text{sgn}(\sigma)\prod_i (PA)_{i,\sigma(i)}$$ $$=\sum_{\sigma\in S_k}\text{sgn}(\sigma)\prod_i \sum_jP_{ij}A_{j,\...
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How can one find the discriminant of a lattice without using a basis?

The setup is a lattice $L$, a finitely generated free abelian group with symmetric positive definite integral bilinear form $\langle \ ,\ \rangle:L\otimes_{\mathbb{Z}} L\rightarrow \mathbb{Z}$, for ...
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2answers
34 views

Relationship Between Determinant and Matrix Rank

Let $n\in \mathbb{N}$, and $S\in \mathbb{R}^{n\times n}$ be a symmetric positive semi-definite (PSD) matrix with rank $r \triangleq \mathrm{rank(S)}\leq n$. Can $r$ be bounded in terms of the ...
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1answer
29 views

Particular cases of Cramer's Rule

While studying Cramer's Rule for non-homogeneous equations in three unknowns one comes across the statement that for the system to possess infinite solutions the following conditions are required: (i)$...
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2answers
396 views

$3 \times 3$ matrix with determinant a large power of $2$

Here's a little curiosity I found. The following $3 \times 3$ matrix has entries that are distinct primes $< 100$ and its determinant is $2^{19}$. $$ \pmatrix{71 & 31 & 97\cr 61 & 67 &...
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1answer
48 views

Finding the determinant of a matrix by using the adjoint

Problem: Find the inverse of the following matrix by finding its adjoint: $$ \begin{bmatrix} -1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix} $$ Answer: The first step is ...
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1answer
35 views

Determinant of A inverse

If A is an invertible matrix of order 2 , then det (A inverse) is equal to A) det (A) B) 1 / (det A) C) 1 D) 0 I tried approaching this question am not getting it but I KNOW ONE ...
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1answer
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Questions in study of Adjoint and inverse in Linear Algebra

While studying Linear Algebra from Hoffman Kunze, I am unable to understand few arguments given in lesson- Determinants. As my Institute is closed, so I have no help other than asking questions here : ...
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117 views

Choosing the sign of determinant when taking a square root

Calculate the determinant $$\det(A)=\begin{vmatrix}a&b&c&d\\ \:\:\:-b&a&d&-c\\ \:\:\:-c&-d&a&b\\ \:\:\:-d&c&-b&a\end{vmatrix}$$ I found that $$\det(A)\...
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1answer
118 views

True or False: If $A$ is square and $|\det(A)|=1$, then $A^{-1}$ has integer entries [closed]

True or False: If $A$ is square and $|\det(A)|=1$, then $A^{-1}$ has integer entries. I spent a good amount of time thinking about this problem. I think the answer would be false as I could see a ...
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1answer
73 views

The sum of these 9! determinants is? [closed]

(image is attached for those who think I have changed the statement of the question while copying from the book) Chose any 9 distinct integers. These 9 integers can be arranged to from 9! determinants ...
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2answers
72 views

$\det(AB) \not = \det(A)\det(B)$? [closed]

What’s wrong with my reasoning? $$\det\left(\left[\begin{matrix} 2&-2\\-2&2\end{matrix}\right]\right)=2\times2-(-2)(-2)=0$$ $$\left[\begin{matrix} 2&-2\\-2&2\end{matrix}\right] \left[\...
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1answer
47 views

Find values that k may take if $k=\det\left(A^3+B^3+C^3\right).\det\left(A+B+C\right)$

Let $A,B,C$ be $n\times n$ matrices with real entries such that their product is pairwise commutative. Also $ABC=O_{n}$. If $$k=\det\left(A^3+B^3+C^3\right).\det\left(A+B+C\right)$$ then find the ...
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34 views

Finding number of solutions?

The number of solutions of the equations $x_{2}-x_{3}=1$ $-x_{1}+2x_{3}=2$ $x_{1}-2 x_{2}=3$ (a) zero (b) one (c) two (d) infinite In a solution,I saw $$ \begin{array}{l} \text { Let } \Delta=\left|\...
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30 views

Prove that determinant of triangular matrix is the product of its diagonal ennltries. [duplicate]

Prove that determinant of triangular matrix is the product of its diagonal entries. I want to use $\operatorname{'sgn'}$.I know that $\operatorname{det}(A)=\sum\limits_{\sigma}A(1,\sigma1)...A(n,\...
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1answer
26 views

For the elementary row operation of exchanging two rows, is it required that the rows be different?

For example, on this page, $i \neq j$ is specified for the elementary row operation of row addition, but not for row switching. Of course, I understand that if $i = j$ and you switch rows $i$ and $j$, ...
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1answer
69 views

Prove that $|\mathbf{A}+\mathbf{B}|=\mathbf{0}$ As per following condition

If $\mathbf{A}$ and $\mathbf{B}$ are real orthogonal matrices of the same order and $|\mathbf{B}|+|\mathbf{A}|=\mathbf{0}$ Prove that $|\mathbf{A}+\mathbf{B}|=\mathbf{0}$ My Approach:- $|\mathrm{A}|+|...
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1answer
17 views

$\det \varphi$ is not a zero divisor $\implies$ Endomorphism $\varphi$ of a free $R$-Module injective

Let $R$ be a commutative Ring with $1$, $M$ a free $R$-Module of rank $2$ and $\varphi \in \operatorname{End}_R (M)$. Show that: If $\det \varphi$ is not a zero divisor in $R$, then $\varphi$ is ...
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2answers
80 views

Closed-form solution for the determinant of a Vandermonde-like matrix

I'm trying to find a closed-form solution $\forall$ odd integer $n\ge 3$ for the determinant of a matrix with some structure on it. After some manipulation, I've reduced it to the following matrix: $\...
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1answer
33 views

A specific change of variable, similar to spherical coordinates

Is it possible to get an explicitly formula for the following change of variable (formula for the Jacobian or for the inverse. I would even accept results from mathematica or other software, which I ...
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80 views

Is it correct this definition of determinant?

Kay, in the beginning of book about tensor calculus, explain Laplace expansion of a determinant. I suppose that (because he doesn't define the determinant of a $2 \times 2$ matrix), he still refers ...
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algebraic branching programs and the 3x3 permanent

Using Grenet's construction (http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=193FC514DC541C17E8F03A7A6BDE4C61?doi=10.1.1.717.4014&rep=rep1&type=pdf ) we can write the 3x3 permanent as ...
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1answer
64 views

Prove that the product of eigenvalues is equal to the determinant

I saw these notes (page 5, question 6) on proving that $$\det(A) = \lambda_1 \cdots \lambda_n$$ where $\lambda_1, \dots, \lambda_n$ are all the eigenvalues of $A \in \mathbb{R}^{n \times n}$. I am ...
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1answer
44 views

Is there any easy way to calculate the value of this determinant?

\begin{vmatrix} 0 & 3 & 1 & 2 & 10! & e^{-7}\\ 1 & 2 & -1 & 2 & \sqrt{2} & 2 \\ -1 & -2 & 3 & -3 & 1 & -\frac{1}{5} \\ -2 & -1 & ...
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1answer
59 views

If $A$ is an orthogonal matrix with $|A|=-1$, show that $|I-A|=0$

Let $A$ be an $n \times n$ orthogonal matrix where $A$ is of even order with $|A|=-1.$ Show that, $|I-A|=0,$ where $I$ denotes the $n \times n$ identity matrix. My approach $A \cdot A^{\top}=I$ $|A| \...
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1answer
110 views

Prove that $\det ((A + B + C) (A^3 + B^3 + C^3-3ABC))\geq 0 $

Suppose that A, B and C are 2x2 matrices that switch between each other. Prove that $$\det ((A + B + C) (A^3 + B^3 + C^3-3ABC))\geq 0. $$ I did $$A^3+B^3+C^3-3ABC=\frac12(A+B+C)((A-B)^2+(A-C)^2+(B-C)^...
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1answer
67 views

Show $\det(F_n)=1$ for all $n$

Consider the $n\times n$ matrix $F_n= (f_{i,j})$ of binomial coefficients $$f_{i,j}=\begin{pmatrix}i-1+j-1\\i-1\end{pmatrix}$$ Prove that $\det(F_n)=1$ for all $n$. My current idea is to apply ...
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1answer
98 views

Calculating the value of a determinant

$\begin{vmatrix} 1 & 2 & 1 & -2 & 1 & 4\\ -3 & 5 & 8 & 4 & -3 & 7 \\ 2 & 2 & 2 & -1 & -1 & -1 \\ 1 & 2 & 3 & 4 & 5 & ...
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25 views

Principal Vector Exercise from Barrett O'Neill textbook

Prove that $\left|\begin{array}{ccc}{{v}_{2}}^{2}& -{v}_{1}{v}_{2}& {{v}_{1}}^{2}\\ E& F& G\\ L& M& N\end{array}\right|=0$ then $\overset{\to }{v}$ is principal vector where ...
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27 views

Determinant of product of matrix and vectors

In determinants, we have the property that $det(AB)=det(A)det(B)$, I believe this can also be extended to the product of three matrices i.e. $det(ABC)=det(A)det(B)det(C)$. Given $X$ is a vector of ...
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0answers
40 views

Determinant of family of Toeplitz matrices. Can we use recursion?

When investigating another question regarding matrix let us call it $M_{10}$ I found a peculiar pattern which I can't prove. We can define $M_n$ to be the $n\times n$ Toeplitz matrix where the ...
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2answers
145 views

Determinant of a Toeplitz matrix

How can I calculate the determinant of the following Toeplitz matrix? \begin{bmatrix} 1&2&3&4&5&6&7&8&9&10\\ 2&1&2&3&4&5&6&7&8&9 ...
4
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4answers
94 views

$\det(I+A)=1+\operatorname{Tr}(A)$ if $\operatorname{rank}(A)=1$.

Let $A$ be a complex matrix of rank $1$. Show that $$\det (I+A) = 1 + \operatorname{Tr}(A)$$ where $\det(X)$ denotes the determinant of $X$ and $\operatorname{Tr}(X)$ denotes the trace of $X$. Any ...
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31 views

space of diagonal matrices with entries $e^{e^{s_1}},e^{e^{s_2}},\cdot\cdot\cdot$ s.t. $\sum_{i \ge1} s_i=0.$

Consider a diagonal matrix $A$ with entries $e^{s_1},e^{s_2},\cdot\cdot\cdot$ on the diagonal s.t. $\sum_{i \ge1} s_i=0.$ From what I understand this is a subgroup of $SL_n(\Bbb R).$ This is because ...
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2answers
49 views

Calculating determinant of $A^n$ given the matrix $A$

Find if $det(A)=det(A^n)$ for $n>1$. How do I tackle questions like this, in general if the matrix $A$ is provided in the question? Should I work out with the basic definition of a determinant, ...

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