Questions tagged [determinant]

Questions about determinants: their computation or their 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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137 views

An identity involving the determinant of a $N\times N$ matrix constructed from N numbers

Given a set of $D$ complex numbers $\{x_1,\dots x_D\}$ such that $x_ix_j\neq 1$ for $i,j\in (1,\dots, D)$, construct a $D\times D$ matrix $Z$ whose matrix elements are given by $Z_{ij}=\frac{1-x^2_i}{...
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64 views

How to determine the eigenvalue of a $3\times 3$ matrix, given the other two complex eigenvalues

Suppose that $A$ is a matrix $3\times 3$ with real entries and such that $\det(A)=1$. Also, suppose that $\lambda_1=(1+\sqrt{3}i)/2$ is a eigenvalue of $A$. The idea is to determine the remaining ...
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31 views

How to prove iff about diagonalizability and determinants?

I have the following linear algebra proof: Question: Let $V$ be an n-dimensional vector space, $\lambda \in \mathbb{R}$, and $T : V \rightarrow V$ a linear map with det($T − tI$) = ($\lambda − t)^n$. ...
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36 views

What are the proprites of identity matrix in $2\times 2$ $\Bbb C$? [closed]

If we have a $2\times 2$ matrix in $\Bbb C$ which is a product of a sequence of other matrices, and we want to show that the matrix is identity, what are the conditions that I should check with the ...
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37 views

Showing that a certain binary matrix cannot be congruent to the null matrix

I don't get why the following matrix (whose entries belong to the binary field) \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} cannot be congruent to the null matrix ,according to my notes. Can ...
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32 views

Invariant subspace proof involving determinant?

This is a linear algebra textbook question: If $W$ is a $T$-invariant subspace, we denote by $T: W \rightarrow W$ be the map defined by $T_W (v) = T(v)$. (a) Let $V$ be finite-dimensional and $W$ a $T$...
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1answer
41 views

Eigenvalue proof with complicated matrix formula

This is a multipart linear algebra textbook question. Let $A, P_1, P_2, N_1, N_2$ be $n$ × $n$ matrices with real entries such that $$ I_n= (t I_n − A) \left(\frac{N_1}{(t-1)^3} + \frac{N_2}{(t-1)^2} +...
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How to solve large matrix equations involving eigenvalues? [duplicate]

This is a multipart linear algebra textbook question. Let $A, P_1, P_2, N_1, N_2$ be $n$ × $n$ matrices with real entries such that $I_n$ = $(t I_n − A) \big(\frac{N_1}{(t-1)^3} + \frac{N_2}{(t-1)^2} +...
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8 views

Inverse an invertible binary matrix plus a constant scalar

Assume I have a matrix $A$ composed of only $0$'s and $1$'s. It is known that A is invertible. Now, consider a shifted version of this matrix denoted by B. B is formed by adding a constant $C$ to all ...
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55 views

What is the meaning of Lax's definition of orientation of simplexes?

I have some questions regarding simplexes after reading Lax's "Linear Algebra and its Applications", the chapter on determinants. (1) I do not understand his definition of orientation. He ...
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1answer
31 views

Shilov's Linear Algebra - Chapter 1, Problem 9 [duplicate]

Calculate the $n$-th order determinant: $$\Delta= \begin{vmatrix} x&a&a&\ldots&a\\ a&x&a&\ldots&a\\ a&a&x&\ldots&a\\ \cdot&\cdot&\cdot&\...
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49 views

Proving that $\det(B) = 0$ [closed]

There are $v_1, v_2, ... ,v_n$ colums with $n$ real components. There is $n\times n$ matrix $A$ which $v_1, v_2, ... ,v_n$ are its colums: $$A = (v_1|v_2|...|v_n)$$ There is $n\times n$ matrix $B$ ...
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63 views

Prove that $\det(T) = \det(T_{w_{1}})\det(T_{w_{1}})\cdots \det(T_{w_{k}})$ where $W_{i}$ are $T$-invariant subspaces of $V$ - Trouble at last step.

Let $T$ be a linear operator on a finite dimensional vector space $V$ and let $W_{1} \oplus \cdots \oplus W_{k}$ where $W_{i}$ are $T$-invariant subspaces of $V$. Prove that $\det(T) = \det(T_{w_{1}})\...
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29 views

Find all ordered pairs of real numbers $(a,b)$ for which there is a unique $2\times 2$ real symmetric matrix $M$.

Find all pair of real numbers (a,b) for which there is a unique 2x2 real symmetric Matrix $M$ with $\operatorname{tr}(M) = a$ and $\det(M) = b$ What I have already tried is forming a characteristic ...
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1answer
57 views

Integration on a submanifold parametrized as the level set of a submersion

Such integrals are usually defined when the submanifold $\mathcal{N}\subset \mathbb{R}^n$ is given (on a neighborhood $V$ of a point $p$) by an immersion $\iota: \mathbb{R}^k\supset U \longrightarrow ...
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2answers
146 views

Why the inverse of a matrix involves division by the determinant?

I was studying inverse matrix. Suddenly I stumbled on the inverse of 3×3 K. And it involved a division by the determinant (Well, only with numbers it was). And it was also said, about involving a ...
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2answers
70 views

Calculate eigenvalues of $\begin{bmatrix}x & 9\\4 & x\end{bmatrix}$ as a function of $x$

Calculate eigenvalues of $$\begin{bmatrix}x & 9\\4 & x\end{bmatrix}$$ as a function of $x$. $$A = \left|{\begin{bmatrix} x & 9 \\ 4 & x \end{bmatrix} -...
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3answers
92 views

How to compute this determinant

I think adding some rows or column to each other will get a triangular matrix which is easy to compute but i cant see how
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1answer
37 views

Proving block triangular matrix determinant formula with 4x4 matrix [duplicate]

Given the A matrix as follow: $$A = \begin{pmatrix}B & D\\\ 0 & C\end{pmatrix}$$ Where B and C are square matrices. Matrix A is said to be in block (upper) triangular form with the formula for ...
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1answer
46 views

prove the lower limit of Hellinger distance for multivariate Gaussians

For two multivariate Gaussians with covariance matrices $A$ and $B$ (of dimension $n$) and with equal means, the squared Hellinger distance is given by $$ H^2 = 1 - \frac{[\det(A)\det(B)]^\frac{1}{4}}{...
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1answer
43 views

How do i calculate the determinant of this matrix? [closed]

How do i prove this relationship? EDIT: I dont have any direction, the minor technique didnt work
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1answer
49 views

Finding characteristic polynomial for 4x4 matrix

I am trying to find the characteristic polynomial for the following matrix: $$ A = \begin{pmatrix}7&1&2&2\\ 1&4&-1&-1\\ -2&1&5&-1\\ 1&1&2&8 \end{pmatrix}...
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1answer
35 views

How to prove determinant of matrices identity?

I have the following linear algebra practice question: Let $A, B, C, D$ be four $n \times n$ matrices. Show that if $D$ is invertible and $CD = DC$, then $$\det\begin{pmatrix} A & B \\ C & D \...
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1answer
58 views

Consider 24 matrices of order 2×2 [closed]

Considering 24 matrices of the type 2 x 2 which can be obtained by some arrangement of 4 non-negative integers a, b, c and d. For certain assignment of non-negative integers, four of these matrices ...
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21 views

Is there a name for “multiplicatively skew-symmetric matrices” and what are good techniques for computing their determinant?

In the course of a counting problem related to graph paths, I encounter a type of matrix that satisfies the following properties: All diagonal elements are zero, that is, $a_{ii} = 0$ for all i ...
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1answer
71 views

Bounds for $\det(AA^T) \ge \det(ABA^T)$

Let $A$ be a $m \times n$ matrix with real entries, and let $B$ be a $n \times n$ real symmetric matrix with absolute eigenvalues $\le 1$. Are there (ideally sharp) bounds for the inequality $$\det(AA^...
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52 views

Approximate the following determinant

Let $f(x)$ be a smooth odd function $f(x) = -f(-x)$ for $x \in \mathbb{R}$. Then I am interested in approximating the following determinant for small $\epsilon$. $$D=\det_{1 \leq i , j \leq n} f((x_i -...
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1answer
34 views

LADW Book Determinants: author claims that linearity can be deduced from scalability and invariance under “column replacement”

When developing theory of determinants in his well known LADW book (https://sites.google.com/a/brown.edu/sergei-treil-homepage/linear-algebra-done-wrong) Treil, prior to even introducing normed spaces ...
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64 views

Determinant of a special kind of matrix

I am learning linear algebra and facing with a problem in my research. I have a special kind of matrix: $$ M_3\left(b_1,b_2,b_3\right)=\left( \begin{array}{ccc} b_1 & a_2 & a_3 \\ -a_1 & ...
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1answer
27 views

Find all $c$ for which a set of vectors is linearly independent set of vectors in $R^3$

Find all $c$ that belongs to $R$ for which $S = \{ (c^2,0,1), (0,c,0), (1,2,1) \}$ is a linearly independent set of vectors in $R^3$. I tried to solve this problem by setting the determinant of these ...
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1answer
34 views

given the det of a matrix find the det of another matrix

$\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}=-4$ Find $\begin{vmatrix} 2a_3 & 2a_2 & 2a_1 \\ b_3-a_3 & b_2-a_2 &...
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1answer
138 views

How to find the sign of the determinant?

Let $A_2=\begin{bmatrix}2^2&1\\3^2&1\end{bmatrix}$ and $A_3=\begin{bmatrix}3^2×2^2&3^2& 1\\4^2×3^2&4^2&1\\5^2×4^2&5^2&1\end{bmatrix}$, and so on. Then, $\det A_2 = 2^2-...
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1answer
79 views

Determinant as the volume of a box in n-dimensions

Below is a proof found in Gilbert Strang's book and here Why is the determinant the volume of a parallelepiped in any dimensions? that the determinant equals the volume of a box : To find the volume ...
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1answer
38 views

What is happening in equation $det(A - \lambda I) = 0$ in basic terms? [duplicate]

When we want to get the eigenvalues of a square matrix $A$ we calculate $$ det(A - \lambda I) = 0$$ What I don't understand is that what we do here in basic terms. Like, for a simple, square matrix M, ...
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1answer
35 views

Determinant of a matrix when its diagonal elements have a certain property

Let $A=\begin{pmatrix} l_1&-1 &-1 &-1&-1 \\ -1&l_2 &-1 & -1 & -1\\ -1 & -1 &l_3 & -1 &-1\\ -1 & -1 & -1 & l_4 & -1\\ -1 & -1 &-1 ...
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25 views

Determinant of the sum of a circulant matirx and a diagonal matrix

Can we compute the determinant of the sum of a circulant matirx and a diagonal matrix in general? In other words, can we compute the determinant of those matrices which have circulant properties but ...
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1answer
69 views

Adjoint of $0, \pm 1$ invertible matrix

Let $A \in \{0, \pm 1\}^{n \times n}$ be an invertible matrix and $B$ be the adjoint matrix of $A$. $A$ is a totally unimodular matrix if its determinant is $\pm 1$ and for all square sub-matrix $C$ ...
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1answer
18 views

Given a matrix A of order $n$, then how many possible order of minor's exists and how to calculate it [duplicate]

Let A be a square matrix of order 3 over $R$ then minor of $a_{11}$ is determinant of the square submatrix of order 2 which is the result of leaving $1st$ row and $1st$ column, like that we calculate ...
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1answer
29 views

Properties of the elements of $SU(n)$

For the group $SU(n)$ I can write its elements in the form: $$ U(n)= \exp(-in^a \lambda_a)$$ Considering $n^a$ to be very small I write: $$ dU = \exp(-i dn^a \lambda_a)$$ and knowing that $dU$ is anti-...
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40 views

How to find $k$ using the determinants matrices properties?

So lets say I want to find $k$ in this question: $\det \begin{pmatrix}2a_1\:&\:2a_2&2a_3\:\\ 3b_1+5c1\:&3\:b_2+5c_2\:&\:3b_3+5c_3\:\\ 7c1\:&\:7c_2\:&\:7c_3\:\end{pmatrix}=k\...
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1answer
75 views

solving determinant with different variable on diagonal but same variable elsewhere?

I'm trying to solve a problem from a textbook. This is a listed problem under the first chapter of the textbook (the book starts with introducing determinant without matrix theory) so the only things ...
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1answer
76 views

using vandermonde determinant to express general determinant?

I am reading an introduction paragraph about Vandermonde determinant and the following statement troubles me: [Statement]: For a determinant $$D_n=\begin{vmatrix} x_{1n}&x_{1,n-1}&...&x_{...
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1answer
83 views

Possible values of a determinant whose entries are $-1$, $0$ and $1$

Let $A$ be a $3\times3$ matrix whose entries in the first column are all equal to $1$. If the other entries are $-1$, $0$ or $1$, then find how many distinct values can $\det A$ take. I want to know ...
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50 views

Intuitive meaning of discriminant as determinant of trace values

Let $A$ be a commutative ring, $n\in\Bbb N$, $R$ be an $A$-algebra which is free $A$-module of rank $n$. If $x_i\in R$ for $i\lt n$, then the discriminant is defined as: $$\operatorname{dis}_{R/A}(x_i:...
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1answer
57 views

A Cauchy type determinant $\det ((x_i+x_j)^{n-1})$

Let $x_1$, $\ldots$, $x_n$ $n$ variables. The $n\times n$ determinant $$ \det \left( (x_i+x_j)^{n-1}\right)$$ is a symmetric homogenous polynomial in $x_1$, $\ldots$, $x_n$ of degree $n(n-1)$, $0$ ...
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1answer
211 views

Proof and Intuition of the Determinant Formula?

[I did notice similar questions were asked here before, but I couldn't find a satisfactory answer for me to grasp as a beginner, so I chose to post this question] I'm just starting to teach myself ...
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22 views

Similarity transform for matrix product with determinant equal to zero.

Context: Consider matrices $A \in \mathbb{R}^{m \times n}$ and $B \in \mathbb{R}^{m \times m}$ where $B$ is invertible. Let where $A$ have $k$ zero columns, so $A$ has the following form: $$A = \begin{...
3
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3answers
164 views

Find rank of very big matrix.

Find rank of $$\begin{bmatrix} 0&1&1\ ..&1&1&1\\ -1&0&1\ ...&1&1&1\\ -1&-1&0\ ...&1&1&1\\ ..&..&..&..&..&..\\ -1&-1&...
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2answers
35 views

Find cofactor matrix of a certain matrix

I'm studying for a qualifying exam in Linear algebra, and I came across this problem: Compute the cofactor matrix of the $n\times n$ matrix $A$ whose elements are $-1$ off-diagonal and $n-1$ on the ...
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16 views

Determinant of the metric tensor for a unit sphere

Consider the following integral, $$ \int_{S^2_R}\frac{1}{2}\varepsilon_{\alpha\beta}\ \varepsilon_{abc}\ \hat{\phi}^a\ \partial_\alpha\hat{\phi}^b\ \partial_\beta\hat{\phi}^c\ \mbox{d}^2\xi $$ where $...

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