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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2
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3answers
96 views

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$?

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$? I read a book a very long time ago where it said something like this: Given a system of two equations with two ...
3
votes
1answer
19 views

Determinant of block matrix with commuting blocks

I know that given a $2N\times 2N$ block matrix with $N\times N$ blocks like $\mathbf{S} = \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ we can calculate ...
1
vote
2answers
30 views

Leibniz Formula, proof of alternating property

$$F_{A} := \sum_{\sigma\in S_n}\operatorname{sign}(\sigma) \prod_{i=1}^n A_{i \sigma(i)}$$ I am trying the prove that $\det(A)=F(A)$. I know that to do this, I need to show that $F$ satisfies the ...
0
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1answer
26 views

determinants of large and infinite matrices

Given a square n x n matrix A, is it possible to find the determinant of the matrix for large values of n easily, and thereby as n goes to infinity? I know that the number of components of the ...
1
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2answers
36 views

Evaluate determinant of an $n \times n$ matrix, help

I need help with this problem: $D_{n}= \begin{vmatrix} 1 & 1 & 0 & \cdots & 0 & 0 & 0 \\ 1 & 1 & 1 & \cdots & 0 & 0 & 0 \\ 0 ...
0
votes
2answers
69 views

Show that $\det(A) > 0$

Let $(a_{ij})$ be a real $n \times n$ matrix satisfying, $a_{ii} > 0 \space (1 \leq i \leq n) ,$ $a_{ij} \leq 0 \space (i \ne j, 1 \leq i,j \leq n) ,$ $\sum_{i=1}^ {i=n} \space ...
2
votes
1answer
49 views

How to evaluate the determinant

How to evaluate this determinant by just using row and column operations ? I'm stuck.Help please! \begin{vmatrix} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & c+b & -2c ...
0
votes
1answer
13 views

Form matrix and calculate it's determinant

I need help with this problem: For every $i,j \in \{1,2,...,n\}$ is $d_{i,j}=min\{i,j\}$. Calculate determinant of a matrix $[d_{i,j}]_{n_Xn}$. Is it right that all the elements of this squared ...
5
votes
4answers
94 views

Prove $\det(I + B) = 2(1 + tr(B)).$

Let A be a $3\times 3$ invertible matrix (with real coefficients) and let $B=A^TA^{-1}$. Prove that \begin{equation*} \det(I + B) = 2(1 + tr(B)). \end{equation*} I know that \begin{equation*} ...
3
votes
1answer
29 views

Matrix with prime entries and largest possible determinant

Let $n\ge 1$ be a natural number. Arrange the first $n^2$ primes in a $n\times n$-matrix, such that the determinant becomes as large as possible. What is the largest possible determinant and which ...
1
vote
3answers
57 views

Determine the number of possible values for $\det(A)$, given that $A$ is an $n \times n$ matrix with real entries such that $A^3 - A^2 -3A +2I=0$.

Determine the number of possible values for $\det(A)$, given that $A$ is an $n \times n$ matrix with real entries such that $A^3 - A^2 -3A +2I=0$. here is the source of the problem. In the last ...
0
votes
1answer
14 views

Determinant of a real skew-symmetric matrix

What will be the value of the determinant of a skew-symmetric matrix of even order when a single element is interchanged between first row and first column? For, $\left| \begin{array}{cccc} 0 ...
1
vote
2answers
43 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
votes
1answer
32 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, ...
3
votes
5answers
118 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 ...
-1
votes
0answers
24 views

For an $n \times n$ matrix A show that $\lambda$ is an eigenvalue for A [closed]

For an $n \times n$ matrix A show that $\lambda$ is an eigenvalue for A if and only if the determinant $det(A - \lambda I) = 0$ where $I$ is the $n \times n$ identity matrix. Can anyone explain this? ...
1
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1answer
26 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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votes
2answers
57 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 $$ ...
0
votes
2answers
36 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
votes
2answers
47 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 = ...
1
vote
3answers
55 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 ...
0
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4answers
41 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
24 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 ...
9
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0answers
103 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 ...
1
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0answers
19 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
20 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 ...
0
votes
1answer
20 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 ...
0
votes
2answers
26 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' ...
-1
votes
0answers
36 views

Find $\det(A)$ where $A_{ij}=1$ if $i\neq j$ and $A_{ij}=3$ if $i=j$. $A$ is of size $n\times n$. [duplicate]

Find $\det(A)$ where $A_{ij}=1$ if $i\neq j$ and $A_{ij}=3$ if $i=j$. $A$ is of size $n\times n$. I already know the answer, but still want to post it here just for your amusement. You may find some ...
1
vote
1answer
19 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 $$\det X_G = \Theta_G.$$ ...
4
votes
1answer
28 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. ...
0
votes
2answers
51 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
votes
3answers
48 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 ...
0
votes
0answers
13 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 $A_i, A_j$ ...
1
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1answer
33 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 ...
1
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1answer
38 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} & ...
1
vote
2answers
33 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 ...
1
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1answer
38 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.
3
votes
1answer
25 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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votes
2answers
63 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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votes
1answer
25 views

Factor of Determinant [closed]

Find the value of $k$ which makes $(x - 2)$ is a factor of the determinant. $$\begin{vmatrix}x-1&x+3&2\\-3&x+5&-6\\x+3&2&x+k\end{vmatrix}$$
0
votes
1answer
40 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
votes
2answers
43 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
votes
1answer
60 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 ...
0
votes
3answers
27 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 ...
2
votes
2answers
49 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
votes
2answers
43 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
votes
3answers
88 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: ...
1
vote
1answer
49 views

Compute the determinant $D_n$

I would like to compute: ...
5
votes
7answers
90 views

For which values of $a,b,c$ is the matrix $A$ invertible?

$A=\begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix}$ ...