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
votes
2answers
192 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
101 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
399 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
1answer
978 views

Invertible matrices over a commutative ring and their determinants

Why is it true that a matrix $A \in \operatorname{Mat}_n(R)$, where $R$ is a commutative ring, is invertible iff its determinant is invertible? Since $\det(A)I_n = A\operatorname{adj}(A) = ...
6
votes
3answers
215 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
5answers
3k views

Show that the area of a triangle is given by this determinant

This is part of my homework. I'm not sure how to start for this question. Can you guys provide some input/hints? Thank you! Let $A=(x_1,y_2)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in ...
6
votes
1answer
646 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
751 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
174 views

Explanation of a cross product result

In my book the result $$(u\times v)\cdot(x\times y)=\begin{vmatrix} u\cdot x & v\cdot x \\u \cdot y & v \cdot y\end{vmatrix},$$ where u, v, x and y are arbitrary vectors, is stated (here ...
6
votes
1answer
182 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
193 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
1answer
127 views

Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} - \det M_{ij}\det M_{ji}$

Let $M$ be a (real) $n \times n$ matrix. For $1 \leq i, j \leq n$ we denote by $M_{ij}$ the $(n-1) \times (n-1)$ matrix that we get when the $i$th row and $j$th column of $M$ are removed. Now, ...
6
votes
2answers
492 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
votes
1answer
566 views

Solving $n$-queens with determinants

I keep reading about a proposed method of finding solutions to the $n$-queens problem using determinants, but I can't find any specific details anywhere. Can somebody explain to me how to find ...
6
votes
1answer
203 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 ...
6
votes
1answer
279 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
votes
1answer
164 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
170 views

Upper Triangular Block Matrix Determinant by induction

We want to prove that: $$\det\begin{pmatrix}A & C \\ 0 & B\\ \end{pmatrix}= \det(A)\operatorname{det}(B),$$ where $A \in M_{m\times m}(R)$, $C \in M_{m\times n}(R)$,$B \in M_{n\times n}(R)$ ...
6
votes
2answers
149 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 ...
6
votes
2answers
222 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 $$ ...
6
votes
2answers
217 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 ...
6
votes
2answers
73 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
163 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
127 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: $$(\det(A-B)+\det(A+B) )^2≥4\det(A^2-B^2 )$$ ...
6
votes
3answers
53 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
489 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
1answer
229 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
3answers
270 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
57 views

Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row?

Let $n\geq2$ be an integer and let $a_1,\ldots,a_n\in\mathbb Z$ with $\gcd(a_1,\ldots,a_n)=1$. Does the equation ...
6
votes
0answers
46 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 ...
6
votes
1answer
128 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
1answer
404 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
6
votes
0answers
236 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 ...
6
votes
1answer
202 views

What does abstract algebra have to say about the determinant?

The determinant is a homomorphism from the multiplicative monoid of matrices to the multiplicative monoid of a field (right?). I find this to be the most intuitive way to interpret some of the ...
6
votes
1answer
155 views

Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
6
votes
0answers
418 views

Determinant of Transpose of Linear Map

I'm trying to find a way to prove that the determinant of the transpose of an endomorphism is the determinant of the original linear map (i.e. det(A) = det(Aᵀ) in matrix language) using Dieudonne's ...
6
votes
1answer
124 views

Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
5
votes
5answers
754 views

Is a square matrix whose diagonal and antidiagonal elements are all zero always singular?

Consider an $n\times n$ matrix whose primary and secondary diagonal elements are all zero. Does it necessarily follow that the determinant vanishes for these matrices? When $n=1,2,3,4$, the matrix is ...
5
votes
5answers
541 views

How to compute the determinant of a tridiagonal matrix with constant diagonals?

How to show that the determinant of the following $(n\times n)$ matrix $$\begin{pmatrix} 5 & 2 & 0 & 0 & 0 & \cdots & 0 \\ 2 & 5 & 2 & 0 & 0 & \cdots & ...
5
votes
3answers
1k views

Is $\det(AB) =\det(BA)$

I am having trouble proving if $$ \det(AB) = \det(BA) $$ is right or wrong. $A,B$ are square matrices. Can you please point me to the right direction? Thank you
5
votes
8answers
1k views

How to calculate the following determinants (all ones, minus $I$)

How do I calculate the determinant of the following $n\times n$ matrices $ \left[ \begin {matrix} 0 & 1 & \ldots & 1 \\ 1 & 0 & \ldots & 1 \\ \vdots & \vdots & ...
5
votes
4answers
374 views

Proof If $AB-I$ Invertible then $BA-I$ invertible.

I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
5
votes
2answers
1k views

Why must the determinant of a matrix with with integer entries be an integer?

Why must the determinant of a matrix with integer entries be an integer? Note: I know what a determinant of a matrix is, not sure how to explain this question. Is that because if the matrix is made ...
5
votes
4answers
214 views

Linear Algebra: different determinant answers

I'm having a problem verifying my answer to this question: Solve for x: $$\left| \begin{array}{cc} x+3 & 2 \\ 1 & x+2 \end{array} \right| = 0$$ I get: $(x+3)(x+2)-2=0$ $(x+3)(x+2)=2$ ...
5
votes
3answers
943 views

Determinant of a linear transformation defining matrix transpose

So if I define a linear transformation $ T: M_{n\times n}(R) \rightarrow M_{n\times n}(R) $ and $ T(A)=A^t $ what would be its determinant?
5
votes
7answers
101 views

For which values of $a,b,c$ is the matrix $A$ invertible?

$A=\begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix}$ ...
5
votes
4answers
361 views

Decompose this matrix as a sum of unit and nilpotent matrix.

Show that the matrix $A=\begin{bmatrix} 1 & 0 \\ 2 & 1 \\ \end{bmatrix}$ can be decomposed as a sum of a unit and nilpotent matrix. Hence evaluate the matrix ...
5
votes
4answers
374 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 ...
5
votes
4answers
631 views

Vector space of polynomials over $\mathbb{R}$ with degree $\leqslant n-1$

Let $P \in \mathbb{R}_{n-1}[X]$ be a polynomial of degree $n-1 \geqslant 0$. Let $\mathbb{R}_{n-1}[X]$ be the vector space of polynomials with degree $\leqslant n-1$ over $\mathbb{R}$. Show ...