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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93 views

Find the eigenvalues for a matrix which is a product of matrices

Suppose I have a matrix $A \in \mathbb{R}^{2, 2}$ which is the product of $3$ other matrices, lets call them $A_1 = \left(\begin{matrix} cosx & -sinx \\ sinx & cos x\end{matrix}\right)$, $A_2= ...
2
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1answer
61 views

Can the terms in a 3x3 determinant be any six nonzero numbers?

Given six nonzero real numbers $x_1,\ldots x_6$, can you construct a 3x3 matrix such that the six diagonal products that appear in the determinant are $x_1,\ldots,x_6$, respectively? In other words, ...
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5answers
142 views

When will $\operatorname{det}\left(A\cdot A^{\top}\right)=0$?

I am writing a small computer program to solve certain linear algebra equations as part of a larger program. For two of my functions I need to evaluate $\left(A\cdot A^{\top}\right)^{-1}$. This got ...
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1answer
65 views

Matrix with unit determinant as a product of elementary matrices.

There are three types of elementary matrices: Type 1: matrices obtained by interchanging the ith row of $I$ and jth row of $I$; Type 2: matrices obtained by multiplying the ith row of $I$ by ...
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2answers
59 views

What are the Eigenvalues of this matrix?

I Need to compute the Eigenvalues of the following General Matrix. Let $b\geqslant a$. Consider the $(a+b+1)\times (a+b+1)$-Matrix $C$ with the following entries. $$ c_{1,1}=c_{a+b+1,0}=1 $$ and $$ ...
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3answers
308 views

Determinant of symmetric matrix $(A-\lambda I)$

If we have a matrix $(A-\lambda I)$ which is: $\left( \begin{array}{ccc} 1-\lambda & -1 & 2 \\ -1 & 1-\lambda & 2 \\ 2 & 2 & 2-\lambda \\ \end{array} \right) $ Then it's ...
2
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2answers
80 views

Proving that the matrix is positive definite

I have looked at similar questions under 'Questions that may already have your answer" and unless I have missed it, I cannot find a similar question. I am trying to answer the following: Let $A = ...
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3answers
606 views

For which values does the Matrix system have a unique solution, infinitely many solutions and no solution?

Given the system: $$\begin{align} & x+3y-3z=4 \\ & y+2z=a \\ & 2x+5y+(a^2-9)z=9 \end{align}$$ For which values of a (if any) does the system have a unique solution, infinitely many ...
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4answers
196 views

Why if the columns of a matrix are not linearly independent the matrix is not invertible?

Why if the columns of a matrix are not linearly independent the matrix is not invertible? I have watched this video about eigenvalues and eigenvectors by Sal from Khan Academy, where he says that for ...
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2answers
37 views

find the smallest interval in which the eigen value of the matrix lie

$$ \begin{bmatrix} 3 & 2 & 2 \\ 2 & 5 & 2 \\ 2 & 2 & 3 \\ \end{bmatrix} $$ I was practicing questions on Matrices & Determinants ...
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0answers
673 views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
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0answers
81 views

Is this proof of the product of determinants in tensor notation correct?

I'll start with the matrix C which is the product of the matrices A and B. $$c^i_k = a^i_jb^j_k$$ The determinant of C is $$\frac{1}{3!}\delta_{ijk}^{rst} c^i_rc^j_sc^k_t $$ by the definition of ...
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0answers
24 views

In which cases is the summation function distributive?

When working on the proof for $$\det(\text{A} -\lambda \text{I})=\det(\text{Q}^{-1}\text{ B Q}-\lambda \text{I})$$ where $\lambda$ is a scalar, $\det$ is the determinant, $\text{I}$ is the identity ...
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1answer
36 views

Mysterious divisibility condition showing up in computation of determinant of certain sparse matrices

Notation: by the $d$'th diagonal of an $n \times n$ matrix $A$ I will denote the diagonal parallel to the main diagonal that starts in row 1, column $d$. I will extend this definition in the obvious ...
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2answers
180 views

Is the Wronskian determinant positive or negative?

The Wronskian of the general solution $y(x)=ay_1(x)+by_2(x)+cy_3(x)$, to a third order differential equation, is given by $$ W = \begin{vmatrix} y_1 & y_2 & y_3 \\ y_1' & y_2' & y_3' ...
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1answer
27 views

Group Determinant Independent of Labeling of Elements

Let $G$ be a finite group with elements $g_1, g_2, \ldots, g_n$. We define the group matrix by $$X_G = [x_{g_ig_j^{-1}}].$$ We then can define the group determinant as $$\det X_G = \Theta_G.$$ ...
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1answer
88 views

Intuition of Wronskian determinant and linear independence

I am wondering the intuition in regard to the following; (let $w$ represent the wronskian function). Please correct me If I am mistaken, but I will write what I do know and what I am confused about. ...
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2answers
59 views

Compute $f_A(\lambda)$ without factoring cubic polynomial?

I'm given the following prompt: "Find the points closest to the origin on the surface defined by $x_1^2+2x_2^2+3x_3^2+x_1x_2+2x_1x_3+3x_2x_3=1$." What's the easiest way to compute the ...
2
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3answers
155 views

Using detA and detB to calculate the determinant of matrix C

If we have C=($A^t$)$^2$BA$^3$B$^-$$^1$A$^-$$^3$ and detA=-2 and detB doesnt equal 0, how do we calculate det C? I know that the transpose of a matrix does not affect the determinant. Does this mean ...
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0answers
17 views

find mean of matrices $A_i, A_j$ given $d_{A_{ji}}=\ln{\left|\left| A_{ji} \right|\right| \left|\left| A_{ji}^{-1} \right|\right|}$

Given a finite set $\mathbb{A}$ of $k$ like-shaped, square, non-singular matrices $A_i\in\mathbb{R}^{n\times n}$, let's define $A_{ji}=A_j A_i^{-1}$, then the distance of the two matrices $A_i, A_j$ ...
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1answer
44 views

Vandermonde determinant and linearly independent (corrected version)

This is a corrected version. Let $a_1,a_2,a_3,b_1,b_2,b_3,b_4,b_5,b_6\in \mathbb{C}$ such that $a_i\not=a_j$ for all $i\not=j.$ If $$\begin{vmatrix} a_1 & a_2& a_3 & b_1 \\ a_1^2 ...
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1answer
83 views

Vandermonde determinant and linearly independent

Let $a_1,a_2,a_3,b_1,b_2,b_3,b_4,b_5,b_6\in \mathbb{C}$ such that $a_i\not=a_j$ for all $i\not=j.$ If $$\begin{vmatrix} a_1 & a_2& a_3 & b_1 \\ a_1^2 & a_2^{2} & a_3^{2} & ...
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2answers
38 views

How to show $\text{rref }[\left.A\right|AB]=[\left.I_n\right|B]$?

For invertible $A^{n\times n}, B^{n\times n}$, how do I show that $\text{rref }[\left.A\right|AB]=[\left.I_n\right|B]?$ Tentatively: $\text{rref ...
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1answer
68 views

What is the determinant of a matrix? [duplicate]

I know how to solve for the determinant of a matrix, but I'm struggling to understand what it represents.
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1answer
152 views

Determinant of skew-hermitian matrix

Given a skew-hermitian matrix $A \in \mathbb{C}^{N\times N}$, then $A = -A^H = -(A^*)^{T}.$ We can also say that $A^T = (-(A^*)^T)^T = -A^*.$ Thus, when computing the determinant we get $$ \det(A) = ...
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3answers
108 views

Determinant properties

Prove without expanding: \begin{equation}\begin{vmatrix}1&1&1\\a^2&b^2&c^2\\a^3&b^3 & c^3\end{vmatrix} = (ab + ac + bc)(b - a)(c - a)(c - b)\end{equation} I tried to zero some ...
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1answer
50 views

Determinant matrix $3 \times 2$

I need to verify the linear dependence or independence of $3 \times 2$ complex matrix, how do I compute the determinant? I would use the row reduced echelon form but I have no idea about how to do ...
2
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2answers
70 views

Matrix identites, Derivative, Determinant, and a kind of Duality involving Traces

I am reading the blog entry Matrix identities as derivatives of determinant identities by Terence Tao, everything is quite clear, up to ~1/3 of the text, where he goes [...] we conclude that $$ ...
2
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1answer
71 views

Closed formula for $\sum_{i=1}^n \det(v_1, \ldots, Av_i, \ldots, v_n)$ [duplicate]

Denote $(v_1, \ldots, v_n)$ the matrix that has columns $v_1,\ldots, v_n\in \mathbb{R}^n$. Let $A\in \mathcal{M}_{n\times n}(\mathbb{R})$. Is there a clever way (without expanding LHS and doing ...
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3answers
29 views

Endomorphism $T$ on $\mathbb{C}^4$ whose charactersitic polynomial is $z(z-1)^3$ but $T(T-I)^2 \neq 0$

Give an example of an endomorphism $T$ on $\mathbb{C}^4$ whose charactersitic polynomial is $z(z-1)^3$ but $T(T-I)^2 \neq 0$ So I'm not very sure how to approach this. I'm guessing T is not ...
3
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2answers
63 views

The positive determinant of one special matrix

I try to prove the positive value of determinant for matrix ($n\times n$ for any $n$): \begin{equation*} ||a_{ij}|| = ||f(x_i - y_j)|| , \text{where}~f = f(\lambda(x - y)) = \exp(-\lambda(x - ...
3
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2answers
83 views

Improve/extend my attempted intuitive explanation for why terms in determinant calculations have alternating signs

The determinant of a shape defined by points $(a,b)$ and $(c,d)$ as labelled in the gif below is $\left|\begin{matrix}a&c\\b&d\end{matrix}\right| = ad-bc$ The following process is the ...
3
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3answers
92 views

Matrix solving equation.

The number of real solutions of equation $$\begin{vmatrix}x^2-12&-18&-5\\10&x^2+2&1\\-2&12&x^2\end{vmatrix}=0$$ is? Well I wanted to do something like this: ...
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1answer
53 views

Compute the determinant $D_n$

I would like to compute: ...
5
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7answers
226 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
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1answer
133 views

Compare determinants of matrices with different dimensions

Reading about matrices and determinants I am wondering about the following concept: How valid is to compare the determinants of matrices with different dimensions? e.g. compare a determinant $D1$ ...
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32 views

Compute det(A) given a function A

Suppose A is a 3×3 matrix and A = 1/3 $u_1\cdot uT_1$ + 1/4 $u_2\cdot uT_2$ + 2/5 $u_3\cdot uT_3$ with $uT_1 = (0, 1, −1)$ $uT_2 = (1, 2, 2)$ $u_3 = (−2,1/2,1/2)$ Compute det(A). I know ...
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1answer
48 views

Determinant Multiplication

I know the following property of the Determinant: $Det(A\cdot B)=Det(A)\cdot Det(B)$ When Trying to prove $Det(Adj(A))=Det(A)^{n-1}$ I came across the following dilemma: $A\cdot Adj(A)=Det(A)\cdot ...
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Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = ...
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0answers
17 views

Determinant over Finite Field [duplicate]

Let there be a matrix $$A_n=\left(\begin{matrix}4&2&\cdots&2\\2&4&\ddots&\vdots\\\vdots&\ddots&\ddots&2 \\2&\cdots&2&4\end{matrix}\right)\in ...
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0answers
56 views

Understanding the formula of the determinant from Shilov's Linear Algebra

I am currently going through Linear Algebra by Shilov and I am having some trouble understanding his derivation of the formula of a determinant. He first introduces the product \begin{equation} ...
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2answers
59 views

Determinant of a square matrix in a field [duplicate]

\begin{array}{rrrrr|r} b & a & a & \cdot \cdot \cdot & a \\ a & b & a & \cdot \cdot \cdot & a \\ a & a & b & \cdot \cdot \cdot & a \\ ...
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0answers
32 views

why is the sum over even and odd permutations the same?

let $m$ be an $n \times n$ matrix (over $\mathbb{R}$,say) and for a permutation $\sigma \in S_n$ define the monomial: $$ P_\sigma(M) = \prod_{j=1}^n m_{j,\sigma(j)} $$ let $\tau$ be an odd ...
2
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81 views

How to prove the following identity?

For any even integer $N$, define two sets $$K_+=\left\{\frac{(2m+1)\pi}{N}|m=-\frac{N}{2},-\frac{N}{2}+1,...,\frac{N}{2}-1\right\}$$ and ...
2
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4answers
409 views

equation of line as a determinant

The question: "Prove that the equation of a line through the distinct points $(a_1,b_1)$ and $(a_2,b_2)$ can be written as $det \begin{bmatrix} x & y & 1 \\ a_1 & b_1 & 1 \\ a_2 & ...
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1answer
42 views

Decompose an invertible $4 \times 4$ real matrix into product of $4 \times 3$ and $3 \times 4$

If we have an invertible matrix $M$ that is $4 \times 4$ and $\left| M \right| \neq 0$ (i.e. it is invertible), is it possible to decompose it into two matrices $4 \times 3$ and $3 \times 4$ ...
2
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2answers
72 views

Determinant of a otherwise constant matrix with a constant diagonal

$$\begin{vmatrix} a-E&-b&-b&-b&-b\\ -b&a-E&-b&-b&-b\\ -b&-b&a-E&-b&-b\\ -b&-b&-b&a-E&-b\\ -b&-b&-b&-b&a-E ...
5
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2answers
121 views

Degree of minimum polynomial at most n without Cayley-Hamilton?

Let $T$ be a linear transformation of an $n$-dimensional vector space $V$ over a field $k$. It's pretty easy to define the minimum polynomial of $T$ and make sure its degree is between $1$ and $n^2$, ...
2
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0answers
77 views

How to compute determinant (or eigenvalues) of this matrix?

Let us have the $n \times n$ circulant matrix given by \begin{equation} C(c_0,c_1,\cdots, c_{n-1}) =\begin{bmatrix} c_0 & c_1 & c_2 &\cdots & c_{n-1}\\ c_{n-1} & c_0 & c_1 ...
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2answers
39 views

Simple question: How do these changes affect the determinant of my matrix?

I assume there's some simple rule to follow, but I can't seem to see what it is. Given $$\det \left[\begin{array}{ccc} a &1 &d\cr b &1 &e\cr c &1 &f\cr \end{array}\right] = ...