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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4
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3answers
279 views

A basic question on determinant and rank of a matrix

How to prove that if the determinant of a $n \times n$ matrix is zero then the rank is less than $n$. I can prove the converse. Only a hint is enough. My definition of rank is the maximum number of ...
4
votes
3answers
156 views

Prove an identity including determinant

Prove that: $$\begin{equation} \begin{vmatrix} x_0^{2n+1}&x_0^{2n}&\cdots&x_0&1\\ x_1^{2n+1}&x_1^{2n}&\cdots&x_1&1\\ ...
4
votes
2answers
75 views

Determinant and trace as conjugations?

For real matrices $A$ it holds that $$\det\,\big(e^A\big)=e^{\mathrm{tr}\,A}$$ so we can write $$\mathrm{tr}=(\exp)^{-1}\circ \;\det\;\circ\;(\exp).$$ Is this interpretation of trace as the ...
4
votes
1answer
196 views

Deriving the formula $\det(AB)=\det(A)\det(B)$ from the geometric property of a determinant

Suppose we are given that the determinant satisfies the following property for any $X\subset\mathbb{R}^n$: $$\widehat{\operatorname{vol}}(\alpha (X))=\det A\cdot\operatorname{vol}(X).$$ Here ...
4
votes
2answers
171 views

Determinant of a matrix with generalized binomial coefficients

Let $$ A= \begin{bmatrix}\binom{-1/2}{1}&\binom{-1/2}{0}&0&0&...&0\\ \binom{-1/2}{2}&\binom{-1/2}{1}&\binom{-1/2}{0}&0&&...\\...&&&\binom{-1/2}{0}\\ ...
4
votes
1answer
103 views

Calculate the determinant

Calculate the determinant $$\begin{align*}D[n]=\begin{array}{cccccc} b & b & b & \dots & b & a \\ b & b & b & \dots & a & b \\ \vdots & \vdots & ...
4
votes
2answers
99 views

Determinants: A Special Condition

Under what conditions is $$ \det(A_1 + \cdots + A_n) = \det(A_1)+\cdots+\det(A_n), $$ just curious.
4
votes
1answer
37 views

Determinant of the matrix $D_n(2,3,1)$

The matrix $D_n(2,3,1)$ is to be written in the form $$\pmatrix{3 & 1 & 0 & 0 & ... & 0 \\ 2 & 3 & 1 & 0 & ... & 0 \\ 0 & 2 & 3 & 1 &... ...
4
votes
1answer
670 views

Unitary matrix proof

Prove that unitary matrix $U$ satisfies $|\det U| = 1$, but $\det U$ is different from $\det U^{H}$. How can I prove these two statements? I guess I should use the fact that every column of unitary ...
4
votes
1answer
226 views

Why is the formula for the determinant as it is?

I know this sounds like such a vague question, but seeing as we all know the formula for the determinant of a general nxn matrix, I want to know exactly why we define it as such. I know that ...
4
votes
1answer
2k views

Are complex determinants for matrices possible and if so, how can they be interpreted?

I've been asked to compute the determinant of a 3x3 matrix with complex entries. I have done so using the normal expansion along a row or column method that I would use were the entries real. My ...
4
votes
1answer
142 views

Cramer's rule: Geometric Interpretation

I have a question concerning Cramer's rule: Let $A$ be a matrix and $A \cdot \vec x = \vec b$ a lineare equation. $A_i$ is the matrix $A$ where the i'th column is replaced by $\vec b$ if $det(A) ...
4
votes
1answer
63 views

Determinants and cofactors?

My professor gave us this definition for determinants for a $n \times n$ matrix $A$: $$\det(A) = a_{11}C_{11} + a_{12}C_{12} ... + a_{1n}C_{1n} $$ where $C_{1j}$ is the cofactor of $A$ on $a_{ij}$. ...
4
votes
2answers
108 views

Determinants of Block Matricies

I read on wikipedia that $Det \begin{pmatrix} A & B\\B& A\end{pmatrix}$ is equal to $ Det(A+B)Det(A-B) $ if $A$ and $B$ commute. Does this hold true even if $ A $ and $ B$ are not ...
4
votes
1answer
881 views

Quick ways to _verify_ determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors …

What are easy and quick ways to verify determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after calculating them? So if I calculated determinant, minimal ...
4
votes
0answers
61 views

Expectation of the absolut value of the determinant of a random matrix

Let $A$ be a random matrix of size $m\times m$ with integer entries $-n\ldots n$. Each value should have the same probability. What is the expectation of the random variable $$X := |\det A|$$ Can ...
4
votes
0answers
67 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 ...
4
votes
1answer
44 views

Maximal determinant of a matrix filled with $\pm 1$

Is there an algorithm to determine what is the maximal determinant you can get just by putting $1$ or $-1$ in a square matrix? For example in a $3\times3$ matrix: $$ \begin{bmatrix}1 && -1 ...
4
votes
0answers
112 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 ...
4
votes
0answers
68 views

The determinant of a special matrix

Recently, I encounter the problem of calculating the determinant of the following matrix $$\left(\begin{array}{cccc} \sin(\theta_1) & \sin(\theta_1 + \delta_1) & \cdots & \sin(\theta_1 + ...
4
votes
0answers
266 views

Understanding a proof about Hilbert Matrix

EDIT: I asked 3 questions. The first one I was able to solve myself, and the other two I cross-posted to MO. Lately I've been interested in the Hilbert Matrix (its definition will come later). I went ...
4
votes
1answer
276 views

Rank of a rectangular Vandermonde Matrix to which weighted columns are added

A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & ...
4
votes
0answers
350 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 ...
3
votes
5answers
2k views

Find the determinant of $I+A$

Let $A$ be a $2\times2$ matrix with real entries such that $A^2=0$.Find the determinant of $I+A$ where $I$ denotes the identity matrix. I proceed in this way :Note that $(I+A)A=A+A^2 \Longrightarrow ...
3
votes
3answers
354 views

Matrix - Show $\det(A) =0$

I am a little stuck on this Matrix problem. Suppose that for complex square matrix A,B the following holds: $AB -BA = A$ Show that $\det(A)=0$ That would mean that A has no inverse. So I thought, ...
3
votes
4answers
226 views

Matrix Determinant

So I'm reading through my linear algebra textbook to review for my final, and happened upon this statement: The determinant of a matrix with positive entries must be positive. Off the top of my ...
3
votes
4answers
168 views

Value of Vandermonde type determinant

Let $x_1,...,x_n $ are distinct real numbers. Is it a formula for the Vandermonde type determinant $V(x_1, \cdots,x_n)$ whose last column is $x_1^k,\ \cdots,\ x_n^k$, where $k \geq n$, instead of ...
3
votes
3answers
242 views

How prove this matrix inequality $\det(B)>0$

Let $A=(a_{ij})_{n\times n}$ such $a_{ij}>0$ and $\det(A)>0$. Defining the matrix $B:=(a_{ij}^{\frac{1}{n}})$, show that $\det(B)>0?$. This problem is from my friend, and I have considered ...
3
votes
4answers
2k views

Prove $\det(kA)=k^n\det A$

Let $A$ be a $n \times n$ invertible matrix, prove $\det(kA)=k^n\det A$. I really don't know where to start. Can someone give me a hint for this proof?
3
votes
2answers
531 views

How to find k given determinant?

So I've got this matrix here, and need to solve for $k$ $$\text{det}\;\begin{pmatrix} 3 & 2 & -1 & 4 \\ 2 & k & 6 & 5 \\ -3& 2 & 1 & 0 \\ 6 & 4 & 2 & ...
3
votes
5answers
146 views

How to find the determinant of this matrix

I have the following matrix: $ \begin{bmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1 \\ 1 & 1 & a & 1 \\ 1 & 1 & 1 & a \\ \end{bmatrix} $ My approach is ...
3
votes
3answers
604 views

How to randomly construct a square full-ranked matrix with low determinant?

How to randomly construct a square (1000*1000) full-ranked matrix with low determinant? I have tried the following method, but it failed. In MATLAB, I just use: n=100; A=randi([0 1], n, n); while ...
3
votes
3answers
311 views

Minimal polynomial, determinants and invertibility

I need to prove: if a matrix $A$ is invertible, then the minimal polynomial $m_a(0) \neq 0$ There is one definition I am unsure of or need help making more clear. I will proceed with proof by ...
3
votes
2answers
955 views

By using properties of determinants show that

$$\begin{vmatrix}1+a^2-b^2&2ab&-2b\\ 2ab&1-a^2+b^2&2a\\ 2b&-2a&1-a^2-b^2\end{vmatrix}=(1+a^2+b^2)^3$$ I have been trying to solve the above determinant. But unfortunately my ...
3
votes
4answers
56 views

Find the product of the following determinants (involving logarithms with different bases)

Find the product of the following determinants: $$\begin{vmatrix} \log_3512 & \log_43 \\ \log_38 & \log_49 \end{vmatrix} * \begin{vmatrix} \log_23 & \log_83 ...
3
votes
2answers
131 views

Calculating determinant of matrix

I have to calculate the determinant of the following matrix: \begin{pmatrix} a&b&c&d\\b&-a&d&-c\\c&-d&-a&b\\d&c&-b&-a \end{pmatrix} Using following ...
3
votes
5answers
459 views

How to find Determinant of a matrix

I could not understand the concept while googling. can anybody provide help? what will be the determinant of the following matrix? $$ \left[\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 5 ...
3
votes
2answers
99 views

Determinant involving recurrence

Evaluate $$\left| A \right| = \left| {\matrix{ {x + y} & {xy} & 0 & \cdots & \cdots & 0 \cr 1 & {x + y} & {xy} & \cdots & \cdots & 0 \cr 0 ...
3
votes
2answers
53 views

The number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$

How to count the number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$? I know that the number of invertible matrices is GL$(n,p)$. Have any ideas?
3
votes
2answers
87 views

Calculating $\det(A+I)$ for matrix $A$ defined by products

Let $b_1,\ldots,b_n\in\mathbb{R}$. I have an $n\times n$ matrix $A$ whose entry is given by $a_{ij}=b_ib_j$, and I'd like to show that $\det(A+I)=\sum_{i=1}^nb_i^2+1$. Define $b=(b_1,\ldots,b_n)$. I ...
3
votes
2answers
182 views

Find matrices $X$ such that for any matrix $Y$ we have $\det(X^2 + Y^2) \geq 0$ [duplicate]

What is the characterization of real matrices $X \in \mathbb{R}^{n\times n}$ such that for any real matrix $Y \in \mathbb{R}^{n\times n}$: $$\det(X^2 + Y^2) \geq 0?$$
3
votes
2answers
79 views

Is this determinant bounded?

Let $D_n$ be the determinant of the $n-1$ by $n-1$ matrix such that the main diagonal entries are $3,4,5,\cdots,n+1$ and other entries being $1$. i.e. $$D_n= \det \begin{pmatrix} ...
3
votes
2answers
119 views

When every term of the determinant is zero

The determinant of $A$ is defined as $$ \det(A) = \sum \pm a_{1,i_1} a_{2,i_2} \cdots a_{n,i_n}$$ Suppose $A$ is a real matrix such that every term in the above sum is zero. Is it true that $A$ has a ...
3
votes
2answers
183 views

How do I prove that the following method to find whether a point lies within a polygon is correct?

I came across the following method to determine whether a given point lies inside a convex polygon - however, I'm not sure how to prove it. Given any three points on the plane $(x_0,y_0)$, ...
3
votes
4answers
185 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 ...
3
votes
2answers
90 views

How find this matrix value of this $\det(A_{ij})$

Find this value $$\det(A_{n\times n})=\begin{vmatrix} 0&a_{1}+a_{2}&a_{1}+a_{3}&\cdots&a_{1}+a_{n}\\ a_{2}+a_{1}&0&a_{2}+a_{3}&\cdots&a_{2}+a_{n}\\ ...
3
votes
3answers
153 views

Is there a way to get all the permutations of $S_4$

I need to calculate the determinant of a $4 \times 4$ matrix by "direct computation", so I thought that means using the formula $$\sum_{\sigma \in S_4} (-1)^{\sigma}a_{1\sigma(1)}\ldots ...
3
votes
2answers
277 views

Finding inverse of a $3\times 4$ or $4\times 3$ matrix

Now I have no problem getting an inverse of a square matrix where you just calculate the matrix of minors, then apply matrix of co-factors and then transpose that and what you get you multiply by the ...
3
votes
2answers
81 views

Minimum of $|\det(X+iC)|$

Let $C$ be a fixed real $n\times n$ matrix, $X$ be an arbitrary real $n\times n$ matrix. Find the minimum value of: $$|\det(X+iC)|=\sqrt{\det(X+iC)\det(X-iC)}$$ When $n=1$ it's clear that the ...
3
votes
2answers
73 views

how to compute the determinant of a linear map

Let $V$ be the vector space of $m\times n$ matrices over a field $F$. Fix an $m\times m$ matrix $A$ and an $n\times n$ matrix $C$, and consider the map $\phi: V\longrightarrow V$ defined by ...