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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6
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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 &...
6
votes
4answers
387 views

Am I misinterpreting this matrix determinant property?

I was reading matrix determinant properties from wikipedia. The property reads $\det(cA) = c^n \det(A)$ for $n \times n$ matrix. However I am not able to realize it. What I find is $\det(cA) = c\det(...
6
votes
2answers
200 views

Why this determinant is conformally invariant?

While I was reading a paper about random analytic function I found a statement that I was not able to prove and after try brute force and search for some references I decided to ask for a help here. ...
6
votes
1answer
111 views

If $A^n = I$, $n$ odd, $A$ a square integer matrix, does $A = I$?

Edit: Crap, even my hypothesis was wrong. If you put $A = \left[ \begin{array}{cc} 1&-1\\3&-2 \end{array} \right]$, then $A^3 = I$ but no eigenvalue is $1$. (What's true is that all ...
6
votes
2answers
428 views

How can I quickly find the determinant of this matrix

$$ \begin{vmatrix} 14 & 2 & 1 & 3\\ 31 & 4 & 5 & 6\\ 26 & 3 & 7 & 4\\ 10 & 1 & 3 & 2\\ \end{vmatrix} ...
6
votes
3answers
524 views

Question about determinants

I am working on some practice problems and I'm unsure where to begin this problem. It starts off by giving $\det(X)= 1$ for the following matrix $X$:$$ \begin{matrix} a & 1 & d \\ b & 1 &...
6
votes
2answers
553 views

Determinant always equal to zero?

I just finished writing a computer program that takes as input a number of matrices and computes the inverse of the product of matrices. To test this program, I wanted to input a 3x2 matrix followed ...
6
votes
5answers
459 views

Calculate the determinant of the $2n \times 2n$ matrix with entries equal to zero on the main diagonal, $1$ below and $-1$ above [duplicate]

Calculate the determinant of the $2n \times 2n$ matrix with entries equal to zero on the main diagonal, equal to $1$ below and equal to $-1$ above. I'll denote this matrix $A_{2n}$. So for example ...
6
votes
3answers
231 views

Formal proof of $\det(I + tA) = \prod\limits_{i=1}^n (1 + t\lambda_i)$

I'm looking for a formal proof for: $$\det(I + tA) = \prod\limits_{i=1}^n (1 + t\lambda_i).$$ I'm very new to matrix theory therefore please forgive me if you find this elementary. Your help in this ...
6
votes
1answer
866 views

Why is the determinant invariant under row and column operations?

I know that we may add any row to any other in a determinant and its value remains the same. This is clear enough since elementary matrices corresponding to row and column operations have determinant ...
6
votes
2answers
891 views

Is the determinant of a zero divisor zero?

Suppose that $A$ is a zero divisor in the ring of $(n\times n)$-matrices over the ring $R$. Is $\det(A) =0$ if $R$ is a field? Is $\det(A) =0$ if $R$ is an integral domain? It's not necessarily ...
6
votes
2answers
86 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
6
votes
1answer
119 views

Problem of determinant when $A^{-1}+B^{-1}=(A+B)^{-1}$

I have two $4\times 4$ real matrices $A$ and $B$, and it is known that $A^{-1}+B^{-1}=(A+B)^{-1}$ ($A$, $B$ and $A+B$ are invertible). How can I prove that $\det (A)=\det (B)$?
6
votes
1answer
2k views

Characterization of positive definite matrix with principal minors

A symmetric matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, ...
6
votes
4answers
883 views

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
6
votes
5answers
120 views

Prove that the determinant is a multiple of $17$ without developing it

Let, matrix is given as : $$D=\begin{bmatrix} 1 & 1 & 9 \\ 1 & 8 & 7 \\ 1 & 5 & 3\end{bmatrix}$$ Prove that the determinant is a multiple of $17$ without developing it? ...
6
votes
1answer
187 views

How to deduce that there are $(n^3+2n-3)/3$ multiplications for the determinant evaluation?

In Friedberg's Linear Algebra, the author points out that the evaluation of the determinant of an $n\times n$ matrix by cofactor expansion along any row requires over $n!$ multiplications, whereas ...
6
votes
4answers
202 views

What is the relation between $\det(A^TA)$ and $\det(AA^T)$?

In the question, $A \in \mathbb R^{m\times n}$ is a matrix, and $\det(\cdot)$ denotes the determinant.
6
votes
2answers
211 views

Prove that determinant of the matrix is non-zero

Given a square matrix $A$ of order $2n$ such that $a_{ii}=0$ and $a_{ij}\in\{-1,1\},\space i\neq j$, prove that $\det(A)\neq0$.
6
votes
1answer
176 views

Area of triangle in determinant form

Area of triangle with vertex $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ is given by : $$\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1\\x_2 & y_2 & 1\\x_3 & y_3 & 1 \end{vmatrix}$$ In this ...
6
votes
1answer
258 views

Determinant of remainder of a primitive matrix modulo 2

I'm trying to prove the following relation for a matrix $A\in \mathbb{Z}^{m\times m} $, $m\geq 2$. It is assumed that the characteristic polynomial of $A$ is primitive modulo $2$: If $C$ is ...
6
votes
2answers
138 views

Degree of minimum polynomial at most n without Cayley-Hamilton?

Let $T$ be a linear transformation of an $n$-dimensional vector space $V$ over a field $k$. It's pretty easy to define the minimum polynomial of $T$ and make sure its degree is between $1$ and $n^2$, ...
6
votes
4answers
201 views

Any hint about solving this monster determinant?

I'm asked to solve the following determinant: $$|A|= \begin{vmatrix} 1 &2 &3 &\cdots &{n-1} &n\\ 2 &3 &4 &\cdots &n &1\\ \vdots &\vdots &\vdots & &...
6
votes
2answers
1k views

What is the maximum possible value of determinant of a matrix whose entries either 0 or 1?

My question is simply the title: What is the maximum possible value of determinant of a matrix whose entries either 0 or 1 ?
6
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1answer
133 views

Determinant of an unknown matrix.

Let $x, y$ be two real variables. If $A$ is any $n\times n$ matrix with all entries in the set $\{x,y\}$ then prove that \begin{equation} \det A = (x-y)^{n-1}(Px + (-1)^{n-1}Qy) \end{equation} where $...
6
votes
1answer
250 views

Determinant of $A + \epsilon X$

In the Wikipedia article on the determinant, it is stated that $$\det \left ( A + \epsilon X \right ) - \det \left ( A \right ) = {\rm tr} \left ( {\rm adj} \left ( A \right ) X \right ) \epsilon + {\...
6
votes
1answer
1k views

Cross product of vectors as a determinant: valid matrix operation?

"The definition of the cross product can also be represented by the determinant of a formal matrix." —Wikipedia This seems like a hack to me—something of much practical use but not ...
6
votes
2answers
10k views

The determinant of adjugate matrix

I have the following proof that I would like to be walked through because I'm not intuitively seeing what to do: If $A$ is $n\times n$, prove $\det\left(\operatorname{adj}(A)\right) = \det(A)^{n-1}$. ...
6
votes
1answer
342 views

Do multiplicative maps of matrices factor through determinants?

Given a map $f:M_n(k)\to k$ (with $k$ some field) such that $f(AB)=f(A)f(B)$ for all matrices $A$ and $B$, is it necessarily the case that $f$ factors through the determinant, i.e. does there exist a ...
6
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1answer
193 views

Evalute big determinant

Today in exam I tried to evaluate this determinant but failed, only somehow "guessed" the answer I got here. Now in home I've managed to find something intuitive, just want to know whether the ...
6
votes
1answer
48 views

Matrix of Ones with Diagonal of Integers

My teacher posed a question to the class today asking us to find the determinant of the following matrix... \begin{bmatrix} 2 & 1 & 1 & 1 & 1 \\ 1 & 3 & 1 & 1 &...
6
votes
2answers
182 views

Sum of squares of maximal minors of a rectangular matrix with orthonormal rows

A matrix $A$ has $m$ rows and $n$ columns, such that $m \leq n$. We know that each row of $A$ has norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is $||x||=\sqrt{x_1^2+x_2^2+......
6
votes
2answers
229 views

Determining the derivation of a determinant

Let $\Phi\colon E\to M$ with $E\subset \mathbb{R}\times M$ and $M\subset\mathbb{R}^n$ open. Consider the function given by $x\mapsto \Phi(t,x)$ for fixed $t\in\mathbb{R}$. (1) Determine $$ \frac{\...
6
votes
1answer
49 views

Determinant of a large block matrix

$\newcommand{\lmt}{\left[\begin{matrix}}$ $\newcommand{\rmt}{\end{matrix}\right]}$ Hi, I was reading through a proof of the number of domino tilings of a $(2n)\times(2n)$ chessboard, and somewhere ...
6
votes
1answer
36 views

Verification for a block-determinant evaluation, and some further thoughts

First, I want some verification for the validity of my approach for this det evaluation question: If $A,B\in M_n(K)$, $K$ is a number field (in the sense that $\Bbb Q$ is the smallest possible one)...
6
votes
2answers
263 views

Determinant of a Certain Block Structured Positive Definite Matrix

PLEASE FIND THE EDITED VERSION OF THIS QUESTION HERE: Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence I WILL ALSO PUT UP A BOUNTY FOR THE EDITED VERSION....
6
votes
2answers
77 views

Order $n^2$ different reals, such that they form a $\mathbb{R^n}$ basis

I've been trying to solve this linear algebra problem: You are given $n^2 > 1$ pairwise different real numbers. Show that it's always possible to construct with them a basis for $\mathbb{R^n}$. ...
6
votes
1answer
327 views

Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case $A = \begin{pmatrix} ...
6
votes
2answers
183 views

Prove: $(\det(A-B)+\det(A+B) )^2 \ge 4\det(A^2-B^2 )$

Let $A,B \in \mathcal{M}_n (\mathbb{R})$ two matrices so that: a) $AB^2=B^2 A$ and $BA^2=A^2 B$ b) $\text{rank}(AB-BA)=1$. Prove: $$(\det(A-B)+\det(A+B) )^2≥4\det(A^2-B^2 )$$ This ...
6
votes
3answers
57 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 &...
6
votes
2answers
742 views

Determinant (and invertibility) of generalized Vandermonde matrix

I have stumbled upon the following generalization of Vandermonde matrix when solving some problem in linear algebra related to Jordan normal form. Let us consider some number $\lambda$ and we assign ...
6
votes
3answers
279 views

Which is easier to work out: determinant or inverse?

Suppose $A\in M_n(R)$ be a $n\times n$ matrix over some ring $R$. Which of the following two tasks is easier? to work out $\det(A)$; to work out $A^{-1}$. More specifically, I want to know the ...
6
votes
1answer
235 views

A particular (functional) determinant calculation

One wants to calculate the quantity, $\det'(\frac{\partial}{\partial t} - i [\alpha, ])$ where the prime on the "det" means that one wants to do a product over only non-zero eigenvalues of the ...
6
votes
2answers
61 views

Find the value of special tridiagonal determinant

Let $A_{n}$ be the following tridiagonal determinant of order $n:$ \begin{vmatrix} a_{0}+a_{1}& a_{1}& 0& 0& \cdots& 0& \quad0\\ a_{1}& a_{1}+a_{2}& a_{2}&...
6
votes
3answers
123 views

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|

I know that B would look something like this: $$\begin{bmatrix} c & a_{12}+c &...&&a_{1n}+c \\ -a_{12}+c & c &...&&a_{2n}+c \\ . \\ . \\ . \\ -a_{1n}+c & -a_{2n}+...
6
votes
2answers
55 views

Help me with the result of this determinant..

$$ D = \begin{vmatrix} 1 & 1 & 1 & \dots & 1 & 1 \\ 2 & 1 & 1 & \dots & 1 & 0 \\ 3 & 1 & 1 & \dots & 0 & 0 \\ \vdots & \vdots & \...
6
votes
1answer
186 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But $||\det(A)|-|\...
6
votes
0answers
82 views

$AB - BA$ is invertible and $A^2 + B^2 = AB \implies 3 | n$ [closed]

Given $A,B \in \mathbb M_n (\mathbb R)$ and that $A^2 + B^2 = AB$ $AB - BA$ has inverse Prove that $3 \mid n$.
6
votes
1answer
140 views

Can we determine the determinant?

Could someone prove that this determinant is not zero? $$\left| \begin{array}{cccc} 1^n & 2^n & \cdots & (n+1)^n \\ 2^n & 3^n & \cdots & (n+2)^n \\ ...
6
votes
0answers
307 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha z_{\bar{\beta}}}...