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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1answer
158 views

Can we say that there exist an integer n such $A+nB$ invertible?

If $A$ and $B$ are $3\times 3$ matrices and $A$ is invertible, then can we say that there exist an integer $n$ such that $A+nB$ invertible? I was trying by choosing n such that eigne values of $A+nB$ ...
4
votes
1answer
59 views

Determinant of sum of matrix with special singular matrix

What is the determinant of the sum of two matrices when one of them is all zeros except for a single column of 1's. I.e. \begin{equation} Det \left[G + S\right] \end{equation} Where \begin{equation} S ...
4
votes
1answer
82 views

Characterization of positive definite matrix with principal minors

A 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, for ...
4
votes
2answers
88 views

Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues

Pretty much what the title suggests: for any positive integer $n$, I'm looking for an $n$-by-$n$ matrix with integer entries, determinant $1$ and $n$ eigenvalues. In case it is absolutely useless to ...
4
votes
1answer
40 views

Formula for determinant of this matrix

Let's have matrix $(n-1) \times (n-1)$ $$ \begin{pmatrix} 3 & 1& 1& \cdots& 1 \\ 1 & 4& 1& \cdots& 1 \\ 1 & 1& 5& \cdots& 1 \\ \vdots &\vdots ...
4
votes
2answers
86 views

Recursive determinant of given matrix in $\mathbb{R}^{n\times n}$

The matrix $A_n\in\mathbb{R}^{n\times n}$ is given by $$\left[a_{i,j}\right] = \left\lbrace\begin{array}{cc} 1 & i=j \\ -j & i = j+1\\ i & i = j-1 \\ 0 & \text{other cases} ...
4
votes
2answers
87 views

$T=-T^{*}$, show that $T+\alpha I$ is invertible.

Please don't answer the question. Just tell me if I am in the right direction. I should be able to solve this. We are given $T=-T^{*}$, show that $T+\alpha I$ is invertibe for all real alphas that ...
4
votes
1answer
73 views

Inequality with determinants problem

Let $A,B \in M_{2}(\mathbb{R})$ with $AB=BA.$ Prove that: $$\det(A^{2}+AB+B^{2})\geq (\det(A)-\det(B))^{2}$$
4
votes
1answer
115 views

Prove that $\det A = 1$ with $A^T M A = M$ and $M = \begin{bmatrix} 0 & I \\ -I &0 \end{bmatrix}$. [duplicate]

Prove that $\det A = 1$ with $A^T M A = M$ and $M = \begin{bmatrix} 0 & I \\ -I &0 \end{bmatrix}$ ($I$ is the identity matrix of order n).
4
votes
3answers
282 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
157 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
47 views

Matrix with entries from $1$ to $16$, each occuring once, and determinant $40800$

In OEIS, it is claimed, that the largest possible determinant of a $4\ x \ 4$-matrix with the entries from $1$ to $16$, each occuring once, is $40800$. Unfortunately, the article does not mention a ...
4
votes
2answers
77 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
38 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
684 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
228 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
146 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
64 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
925 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
1answer
34 views

How do they go from implicit partial differentiation in this problem to solving with a determinant?

In this book I'm studying I've come across a problem where the author solves a partial differentiation problem using determinants. I'm somewhat familiar with them, but I don't see how they derive the ...
4
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0answers
65 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
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
116 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
1answer
55 views

Determinant of a circulant matrix as Chebyshev-like recurrence

It is while studying the Hückel Method of Physical Chemistry that I came across the following recurrence relation: \begin{align*} U_n(x)=xU_{n-1}(x)-U_{n-2}(x)+(-1)^{n-1}(4+2x) \end{align*} Where ...
4
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0answers
69 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
269 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
281 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
351 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
355 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
227 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
170 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
244 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
2answers
146 views

Block matrix determinant

I have encountered an statement several times while proving determinant of a block matrix. $$\det\pmatrix{A&0\\0&D}\; = \det(A)det(D)$$ where $A$ is $k\times k$ and $D$ is $n\times n$ ...
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
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2answers
1k 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
2answers
553 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
147 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
613 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
321 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
4answers
123 views

Looking for an elegant proof of $\det(A) = \det(A^t)$ without Schur decomposition

Looking for an elegant proof of $\det(\textbf{A}) = \det(\textbf{A}^{t})$ without Schur decomposition. Proof 1 with Schur decomposition $$\textbf{A} = \textbf{P}^{t}\Delta\textbf{P} ...
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
134 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
464 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 ...