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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1answer
837 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
3answers
356 views

Computing determinant of a matrix with non-zero values on three diagonals

let $A$ be an $n\times n$ matrix with entries $a_{ij}$ such that $a_{ij}=2$ if $i=j$. $a_{ij}=1$ if $|i-j|=2$ and $a_{ij}=0$ otherwise. compute the determinant of $A$. using the famous formula ...
6
votes
2answers
879 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
83 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
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
5answers
118 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
2answers
208 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
165 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
249 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
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
273 views

Cross products?

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
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
votes
1answer
129 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
248 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
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
335 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
190 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
47 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
177 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
228 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
1answer
35 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 ...
6
votes
2answers
256 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
76 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
295 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
171 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
56 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
694 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
234 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
112 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 & ...
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
0answers
81 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
0answers
153 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
136 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
292 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
270 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
163 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
482 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 ...
5
votes
5answers
923 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
988 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
3k 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
4answers
224 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
2answers
424 views

Determinant called Grammian

Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
5
votes
2answers
120 views

$Tr(A^2)=Tr(A^3)=Tr(A^4)$ then find $Tr(A)$

Let $A$ be a non singular $n\times n$ matrix with all eigenvalues real and $$Tr(A^2)=Tr(A^3)=Tr(A^4).$$Find $Tr(A)$. I considered $2\times 2$ matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ ...
5
votes
3answers
1k 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
237 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
443 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
3answers
175 views

Determinant of a special $n\times n$ matrix [duplicate]

Compute the determinant of the nun matrix: $$ \begin{pmatrix} 2 & 1 & \ldots & 1 \\ 1 & 2 & \ldots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 ...