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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0
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1answer
28 views

Finding out of a set of 3x1 matrices are linearly independent or dependent

I know how to determine if any $2 \times 2$ matrix or $3 \times 3$ matrix is linearly dependent/independent; It's easy, as long as the determinant of the matrix $\ne 0 \implies $ linearly independent, ...
0
votes
1answer
30 views

Extending dimension of matrix to get it determinant. What I'm doing wrong? Or am I right?

Let the matrix of dimension 4 be: $$A=\begin{bmatrix} a11 & a12 & a13 & a14\\ a21 & a22 & a23 & a24\\ a31 & a32 & a33 & a34\\ a41 & a42 & a43 & a44 \...
1
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0answers
41 views

Challenging calculation of a Jacobian for an unusual matrix coordinate transformation

I am studying a random matrix ensemble and I am having trouble performing a coordinate transformation. My question is very straightforward, but perhaps a bit technical. I have the following integral--...
0
votes
1answer
15 views

Area of the region bounded by four vectors.

I'm stuck on how to approach this problem. I have a feeling it involves determinants and linear algebra. It's to find the area of the region bounded by the vectors: [-7,7], [5,5], [3, -4], [-5,-6]
1
vote
2answers
31 views

Finding the limit of a Matrices determinant

The problem is as follows: I've been trying to figure this out with no luck. I'm lost at the $A_k+1$ and $A_0$. I'm not sure what they are implying and how they would apply in finding the limit.
2
votes
3answers
42 views

Is the Gramian determinant always nonnegative?

Is Gramian determinant $\det (A^TA)$ always nonnegative (or at least when $A$ has no more columns than rows)? It's used to compute a volume element as in this article https://en.wikipedia.org/wiki/...
3
votes
1answer
64 views

Binary matrices with rank $n$

I'm stuck doing this problem Let $A$ be a matrix of order $n \times n$ with entries in $\{0,1\}$, which has exactly two $1$'s on each row and on each column. Which conditions are necessary and ...
-2
votes
2answers
112 views

Is a correlation matrix with positive determinant PSD?

Please note: I'm not interested in the difference between positive definiteness and semi-definiteness for this question. A correlation matrix is a symmetric positive semi-definite matrix with 1s down ...
6
votes
4answers
193 views
+50

Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$

Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that: $$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$ under the following ...
7
votes
2answers
95 views

If GCD $(a_1,\ldots, a_n)=1$ then there's a matrix in $SL_n(\mathbb{Z})$ with first row $(a_1,\ldots, a_n)$

Since the gcd of the integers $a_1,\ldots, a_n$ is $1$, there exists weights $x_i \in \mathbb{Z}$ such that $a_1x_1+\cdots+ a_nx_n=1$. My two ideas are (a) to brute force construct an $n\times n$ ...
1
vote
1answer
40 views

Existence of matrices with non-zero principal minors

The problem sounds very simple but I have yet to come to an answer. Prove or disprove: For all $n$ there exists a matrix $A \in \mathbb{R}^{n \times n}$ with $\det(A) = 0$ such that all first ...
1
vote
3answers
34 views

Properties of RREF 3x3 matrix is the identity

The row reduced echelon form of a 3 × 3 matrix A is the identity. State whether each of the following is true or false. You do not need to explain your answers. (a) A has an inverse. (b) The columns ...
1
vote
1answer
33 views

Help with proving a 2 by 2 determinant is the area of parallelogram

I have proved a large part of this by the following but get stuck at the last step. To say $A=ad-bc$, we still need $ad>bc$. I have puzzling over this for hours. Thank you!
1
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0answers
26 views

Calculating determinants

Let $n\geq 2$ be an integer and let $\Sigma$ be the collection of all $2$-subsets (a 2-set is a set that contains $2$ elements) of $[n]=\{1,2,\dots,n\}$, thus $\Sigma$ contains $\binom{n}{2}$ elements....
6
votes
4answers
202 views

What is the relation between $\det(A^TA)$ and $\det(AA^T)$?

In the question, $A \in \mathbb R^{m\times n}$ is a matrix, and $\det(\cdot)$ denotes the determinant.
40
votes
5answers
2k views

Why is the determinant defined in terms of permutations?

Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of ...
1
vote
1answer
47 views

Expansion of a determinant in powers of $\lambda$ (from Courant-Hilbert Vol I)

On page 20-21 of volume I of Courant & Hilbert's "Methods of Mathematical Physics" they say: If we expand the determinants $\Delta(u,y;\lambda)$ and $\Delta(\lambda)$ in powers of $\lambda$, ...
-1
votes
3answers
89 views

Relationship between 2 determinants [closed]

Let $D_1= \begin{vmatrix}a_1 & b_1 & c_1\\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{vmatrix}$ and $D_2= \begin{vmatrix}a_1+pb_1 & b_1+qc_1 & c_1+ra_1\\ a_2+pb_2 &...
0
votes
2answers
58 views

Complex numbers inside determinant

Let $ \begin{vmatrix}6\iota & -3\iota & 1\\ 4 & 3\iota & -1\\ 20 & 3 & \iota \\ \end{vmatrix}= x +\iota y$, then what are the values of $x$ and $y$?
2
votes
1answer
74 views

Determinant of a tridiagonal matrix with a superdiagonal of ones and a subdiagonal of minus ones

$$ D_n = \begin{vmatrix} a_1 & 1 & 0 & \cdots& 0 & 0\\ -1& a_2 & 1 & \cdots & 0 & 0 \\ 0 & -1 & a_3 & \cdots & 0 & 0 \\ \vdots & \vdots &...
1
vote
0answers
41 views

Determinant of a block rectangular matrix

Assume that $a \in \mathbb{R}^{n \times 1}, b \in \mathbb{R}, P \in \mathbb{R}^{n \times n}, u \in \mathbb{R}^{n \times 1}, x \in \mathbb{R}^{n \times 1}$ and $\lambda \in \mathbb{R}$. Now I want to ...
0
votes
1answer
45 views

Proving Determinants equal to some expressions(tips and tricks) [on hold]

Is there any trick or a step by step method that I can use to prove a certain determinant equal to a complicated expression by solving it? Also when I try to open the determinant to prove it equal to ...
-1
votes
2answers
48 views

Condition check for matrices

If a matrix $A= \begin{bmatrix}2a & 2b \\ 2c & 0 \\ \end{bmatrix} $ and matrix $B=2 \begin{bmatrix}a & b \\ c & 0 \\ \end{bmatrix} $, then how is $A=2B$ also explain how is this ...
1
vote
1answer
62 views

determinant of TS, where T is rotation and S is reflection operator

Let T and S be linear transformations from $\mathbb R^2 \to \mathbb R^2.$Let T rotate each vector counter clockwise through an angle $\theta$ about origin and let S be the reflection about the line $y=...
6
votes
2answers
61 views

Find the value of special tridiagonal determinant

Let $A_{n}$ be the following tridiagonal determinant of order $n:$ \begin{vmatrix} a_{0}+a_{1}& a_{1}& 0& 0& \cdots& 0& \quad0\\ a_{1}& a_{1}+a_{2}& a_{2}&...
4
votes
0answers
64 views

Quadrics intersecting the twisted cubic and a line.

I am trying to understand the determinantal approach on Harris book "Algebraic Geometry: A first course" on proving that the intersection of two quadrics containing the twisted cubic in $\mathbb{P}^3$ ...
1
vote
1answer
52 views

Calculate the determinant of $\det(5(AB^{-2})^T)$

I have a matrix $$A = \begin{pmatrix} 0 & 0 & −2 & −7\\ 2 & 2 & 0 & 0\\ 0 & 0 & 1 & 3\\ 5 & 6 & 0 & 0\\ \end{pmatrix}$$ ...
0
votes
2answers
68 views

Eigenvalues of a $3\times 3$ symmetric matrix [duplicate]

Given a $3\times 3$ symmetric matrix \begin{equation} M= \begin{pmatrix} A & B & C \\ B & D & E \\ C & E & F\\ \end{pmatrix}, \end{equation} how do I find the eigenvalues? ...
4
votes
5answers
93 views

Prove that the determinant is $(a-b)(b-c)(c-a)(a+b+c)$

I have the determinant : \begin{vmatrix} 1 &1 &1 \\ a &b &c \\ a^3 &b^3 &c^3 \\ \end{vmatrix} How do I prove that this determinant is equal to $$ (a-b)(b-c)(c-a)(a+b+c) $$
0
votes
0answers
18 views

Stationary distribution of finite-state Markov chain in terms of determinants/products of eigenvalues

I have an $M$-state continuous-time Markov chain with transition-rate matrix $K$ (the column sums are zero), which has $M$ distinct eigenvalues $\lambda_i$, $i=1,\dots,M$. $\lambda_M=0$, so $K$ has ...
4
votes
2answers
82 views

Determinant of $A$

I am trying to solve the following problem: Let $$A^2=\begin{bmatrix} -2 & 2 & -4 \\ 2& 1 & -2\\ 4 &-6 & 6 \end{bmatrix}$$ Consider the trace of the matrix $A$ is $-1$. ...
1
vote
2answers
78 views

A possible generalized determinant?

This will likely seem a bit contrived, and admittedly it is, but I wanted to see just how "close" we could get to generalizing the concept of a determinant. In what follows, we will lose quite a few ...
2
votes
0answers
61 views

Determinant of a block matrix $2n$ by $2n$

Consider the block $2n \times 2n$ matrix $$\begin{bmatrix} A&B\\ 0&D \end{bmatrix}$$ where $A,B,D$ are $n \times n$ blocks. Show that $$\det\begin{bmatrix} A&B\\ 0&D \...
0
votes
3answers
132 views

Show the determinant of an identity matrix multiplied by a vector is equal to an element of the vector

I'm working out a few exercises for an exam, this is an interesting problem that should be simple (about 2 marks) but I can't seem to wrap my head around it. The question is: Let $I$ be the $3\times ...
1
vote
1answer
32 views

Are there geometric representations for determinants? [duplicate]

For whatever reason, my mind wants to organize determinants and other matrix-related things (like bases) geometrically. But I can't wrap my mind around how that would be possible. Is there actually a ...
1
vote
0answers
51 views

How to compute the given determinant?

This is in association with this problem:How to compute $\det(A+J)$? As I could not find suitable answers so I am posing this question with my following attempt: On expanding $\det(A+J)$ by Laplace ...
2
votes
1answer
19 views

Jacobian determinant of a map?

For $m,n\in \mathbb N$, let $f$ is the map given by $$\begin{align} f: & \quad \mathbb R^m \times \mathbb R^n \longrightarrow \mathbb R^m \times \mathbb R^n \\ & (x,y)\mapsto f(x,y) = (x+x',...
1
vote
3answers
62 views

What is the rank(AB) and rank(BA)?

Let $A$ and $B$ be two $n\times n$ matrices such that rank($A$) $=n$ and rank($B$) $=n-1$. Then I know that, rank($AB$) $=$ rank($BA$) $\leq$ min{ rank($A$), rank($B$)} $=n-1$ My question : Is ...
2
votes
1answer
45 views

What is the exact relation between a full rank matrix and its determinant?

I would like to know in particular whether a full rank matrix necessarilly has a determinant equal to/different from zero. Also, would the answer change based on the matrix being squared or not?
0
votes
2answers
41 views

Finding other eigenvalue of matrix given one eigenvalue

I have two questions: 1) Suppose $A$ is a diagonalizable $2 \times 2$ matrix and has determinant $1$. Suppose also that one of the eigenvalues of $A$ is $2$. Find all the eigenvalues of $A^{-1}$. 2) ...
3
votes
3answers
70 views

Let A be a square matrix of order $3$ with integer entries such that $\det(A)=1$.

What is the maximum possible number of entries of A that are even? What if A is a matrix of order n? What is the maximum possible number of entries of A that are PRIMES? What if A is a matrix of ...
0
votes
1answer
25 views

Calculate the determinant of the matrix using cofactor expansion along the first row

The problem: A block diagonal matrix is a square matrix where nonzero element occurs in blocks along the diagonal. an example of a 4x4 block diagonal matrix with two 2x2 blocks is $$ A_{}...
0
votes
1answer
49 views

Minors of a Vandermonde matrix

I am working with the $n$x$n$ Vandermonde matrix where the "$\alpha_i$'s" form the set of integers from 1 to $n$. That is entry $a_{ij} = i^{j-1}$, What I would like to know is if I delete an equal ...
3
votes
1answer
81 views

How to compute $\det(A+J)$?

If $A$ be an $n\times n$ matrix and $J$ be a matrix of same order with all entries $1$ then Show that $\det( A + J)=\det A$ + sum of all cofactors of $A$. I have tried using Laplace Expansion ...
3
votes
1answer
45 views

Similarity classes of matrices

Let $M_n(K)$ be the set of all $n\times n$ matrices over a field $K$. If $\mathcal{R}$ is the equivalence relation defined by matrix similarity, what does the quotient $M_n(K)/\mathcal{R}$ looks like? ...
4
votes
1answer
46 views

Given block matrix $M$, show determinant relationship between $M$ and the block elements of $M.$

Given that $M = \begin{pmatrix} A & B \\ C &D \end{pmatrix}$ and $M^{-1} = \begin{pmatrix} P & Q \\ R & S \end{pmatrix},$ where $A, B,\dots$ are $k \times k$ matrices, show that $\det(...
1
vote
1answer
52 views

Expected number of times to get arbitrary arrangement of coins

I'm thinking about a question: We consider tossing coins repeatedly. Using $+1$ to denote front and $-1$ back, given a positive interger $m$ and $\sigma=(\sigma_1,\dots,\sigma_m)$ where $\sigma_i\in\...
9
votes
5answers
163 views

Compute $\det{T}$ where $T(X)=AX+XA$

Consider the linear transformation $T:V\to V$ given by $T(X) = AX + XA$, where $$A = \begin{pmatrix}1&1&0\\0&2&0\\0&0&-1 \end{pmatrix}.$$ Compute the determinant $\det T$. ...
1
vote
1answer
29 views

Determinant comparison about skew-symmetric matrices

Suppose $S$ is a real skew-symmetric matrix, show that $\det(I+S) \geq 1$, where equality holds iff $S=0$. My idea is to define a function $f(t)=\det(I+tS)$, for a fixed $S \neq 0$, and then show ...
0
votes
1answer
42 views

Determinant of $e^A$. [duplicate]

Suppose $A$ is a matrix in $\mathbb{R}^{k \times k}$. Show that $$\det e^A = e^{\text{Tr } A},$$ where $\text{Tr } A$ is the trace of the matrix $A$. This is marked as a "starred exercise" in ...