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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3
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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
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1answer
49 views

Compute the determinant $D_n$

I would like to compute: ...
5
votes
7answers
96 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
votes
1answer
65 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 $D1$ ...
0
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0answers
29 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 ...
1
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1answer
30 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 ...
30
votes
2answers
829 views

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) = ...
0
votes
0answers
16 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 ...
0
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0answers
35 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} ...
1
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2answers
39 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 \\ ...
2
votes
0answers
21 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
votes
0answers
76 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
votes
4answers
49 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 & ...
1
vote
1answer
36 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
votes
2answers
56 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 ...
2
votes
1answer
54 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
votes
0answers
44 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 ...
1
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2answers
34 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] = ...
1
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0answers
44 views

Path-components of the general linear group using only elementary algebra

Let $E(c)$ be an elementary matrix of the type to add $c$ times a row to another row when applied to another matrix on the left (with $c$ in some off-diagonal position $(i, j)$), and, with the usual ...
0
votes
1answer
25 views

Sum-of-divisors determinant

Let $\sigma_k(n)=\sum_{d|n}d^k$ be the generalized sum-of-divisors function. Let $S_n$ be the matrix defined by $[S_n]_{ij}=\sigma_i(j)$. I read a comment somewhere that $$\det(S_n)=1!\cdot 2!\cdots ...
2
votes
1answer
47 views

Proving Cayley formula using Kirchhoff matrix theorem?

To count the number of spanning trees of a complete graph of order $n$ one can use Kirchhoff matrix theorem and arrive at the exact answer $n^{n-2}$. But in doing so, one should know how to evaluate ...
2
votes
1answer
60 views

Use determinants to calculate the area bounded by 3 vectors

I have seen the proof of why the area of the parallelogram created by 2 vectors $u = \left(\begin{matrix} u_1\\ u_2 \end{matrix}\right)$ and $v = \left(\begin{matrix}v_1 \\ v_2 \end{matrix}\right)$ ...
1
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0answers
26 views

Unknown functions yield a given determinant

I am trying to develop a nomogram which simultaneously shows the exact Fisher equation $(1+u) = (1+v)(1+w)$ and its linear approximation $u \approx v + w$. This amounts to finding twelve smooth ...
2
votes
3answers
119 views

How can we memorize the formula for the determinant of a $4\times4$ matrix?

This is the formula for the determinant of a $4\times4$ matrix. . 0,0 | 1,0 | 2,0 | 3,0 0,1 | 1,1 | 2,1 | 3,1 0,2 | 1,2 | 2,2 | 3,2 0,3 | 1,3 | 2,3 | 3,3 . ...
1
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3answers
71 views

Determinants of $3A, -A, A^2, A^{-1}$, where A is an $4\times 4$ matrix and $\det(A) = \frac{1}{3}$?

How do I find the determinants of the matrices $3A, -A, A^2$, and $A^{-1}$, where $A$ is a $4\times4$ matrix and $\det(A) = \frac{1}{3}$? I know that to find the determinant we can use elimination ...
5
votes
4answers
367 views

Am I misinterpreting this matrix determinant property?

I was reading matrix determinant properties from wikipedia. The property reads $\det(cA) = c^n \det(A)$ for $n \times n$ matrix. However I am not able to realize it. What I find is $\det(cA) = ...
3
votes
1answer
63 views

Verifying whether a number is the determinant of a matrix

What is the (computationally) fastest way to determine whether a number is the determinant of a given real matrix? I am wondering if I have an upper bound on the absolute value of the determinant of ...
1
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0answers
22 views

How to find the determinant of a parallelogram using the vertices. (Using a matrix)

What it says on the tin. I already know how to find the determinant of a parallelogram using the vector components in a matrix, however, I am curious if there is a way to do it simply through the ...
9
votes
3answers
346 views

Calculate a determinant.

Let $a_{1}, \cdots, a_{n}$ and $b$ be real numbers. I like to know the determinant of the matrix $$\det\begin{pmatrix} a_{1}+b & b & \cdots & b \\ b & a_{2}+b & \cdots & b ...
1
vote
2answers
60 views

Can the given transformation possible for given determinant?

In forth step $(x-1)(x-2)$ is obtained by applying transformation R$1 \frac{1}{(x+1)}$ and R$2 \frac{1}{(x+2)}$. But we get value of $x = -1$ or $ x = -2$ so $\frac{1}{(x+1)}$ and ...
3
votes
1answer
37 views

Evaulate a determinant involving factorials.

In a problem set given by a teacher, there is the following problem. If $a_n = \frac{1}{n!}$, evaluate $$ D_n = \begin{vmatrix} a_1 & a_0 & 0 & 0 & \cdots & 0 & 0 & ...
0
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0answers
18 views

Determinant expansion about rows

Here is a theorem from Beezer's A First Course in Linear Algebra: I am having a hard time understanding why there is a $-1$ in $i - 1 + l - \epsilon_{lj}$, could anyone explain?
4
votes
2answers
67 views

If $BA$ has $-1$ as an eigenvalue, then so does $AB$?

I was just encountered with a rather tough problem as follows: Suppose $A,B\in M_n(\mathbb R)$, prove: $$\det(I_n+AB)\ne0\Rightarrow\det(I_n+BA)\ne0$$ Although at this moment I am still at a ...
8
votes
3answers
2k views

Determinant of a 5 × 5 matrix

I have a little problem with a determinant. Let $A = (a_{ij}) \in \mathbb{R}^{(n, n)}, n \ge 4$ with $$a_{ij} = \begin{cases} x \quad \mbox{for } \,i = 2, \,\, j \ge 4,\\ d \quad \mbox{for } ...
5
votes
1answer
109 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
1
vote
1answer
69 views

Tutte matrix - Determinant

I'm trying to understand the proof of the "magic theorem" about the Tutte matrix which states: Let $T$ be the Tutte matrix of $G(V, E)$. Then, $$\det(T) = 0 \quad\Longleftrightarrow\quad G ...
1
vote
1answer
62 views

Finding determinant of a 4x4 matrix

I am trying to find the determinant of this matrix but was told by my teacher that we wouldn't need to find the determinant of more than $3\times 3$ matrices so I am guessing there is a way of solving ...
0
votes
1answer
19 views

Determinant with one parameter, how to deal with this?

Let $t\in \mathbb R$ be a parameter, and $$|A(t)|= \begin{vmatrix} a_{11}+t &a_{12}+t &\cdots &a_{1n}+t\\ a_{21}+t &a_{22}+t &\cdots &a_{2n}+t\\ \vdots &\vdots ...
0
votes
2answers
48 views

Finding complex eigenvalues

For the matrix \begin{pmatrix}1/2 & 1 & 3/4\\2/3 & 0 & 0\\0 & 1/3 & 0\end{pmatrix} Find the eigenvalues and corresponding eigenvectors. I did this with an online calculator and ...
0
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3answers
40 views

Why would the Jacobian not be zero in this case?

Find the jacobian of the transformation x = u, y = 3uv in the uv plane. Why would $U_y$ not be zero in this case, if the equation U = x contains no mentions of y?
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0answers
28 views

How to find spectral radius of ${0,1}$ and ${0,1,-1}$ matrices?

[this is kind of a continuation of this question ] It seems that the following is true, Among $n=3$ dimension symmetric matrices over $\{0,1\}$ which have $d=7$ ones the maximum spectral radius is ...
1
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2answers
75 views

An (open?) problem about a sequence of nested sub-matrices and their determinant

I had an idea. Let us start with an example. Consider the matrix $$ A = \left[ \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right] $$ It is invertible, ...
0
votes
2answers
65 views

Process of finding the eigenvalues of a 3x3 matrix

I'm trying to find the eigenvalues of a 3x3 matrix in order to eventually find an orthogonal matrix $Q$ and diagonal matrix $D$ such that $Q^TAQ = D$, where $A$ is a symmetric matrix, however I'm not ...
6
votes
4answers
193 views

Any hint about solving this monster determinant?

I'm asked to solve the following determinant: $$|A|= \begin{vmatrix} 1 &2 &3 &\cdots &{n-1} &n\\ 2 &3 &4 &\cdots &n &1\\ \vdots &\vdots &\vdots & ...
1
vote
2answers
36 views

Prove that $\det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, (A+uv)^{-1}$.

Let $A$ be a $n \times n$ matrix, $u$ a $n \times 1$ matrix and $v$ a $1 \times n$ matrix. If $A$ and $(A+uv)$ are invertible, prove that $$ \det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, ...
0
votes
1answer
59 views

What is the determinant of the sum of a diagonal matrix and a matrix of ones?

Given a square matrix, all elements outside of the main diagonal being equal to $1,$ what is its determinant?
0
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0answers
9 views

Link between the cofactors of two related symmetric positive-definite matrices

Let $S = \left( s_{ij} \right)_{i,j\in\left\{1...d\right\}^2}$ be a symmetric positive-definite matrix. Let $\Sigma = \left( \sigma_{ij} \right)_{i,j\in\left\{1...d\right\}^2}$ be a symmetric ...
1
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2answers
51 views

Induction for Vandermonde Matrix

Given real numbers $x_1<x_2<\cdots<x_n$, define the Vandermonde matrix by $V=(V_{ij}) = (x^j_i)$. That is, $$V = > \left(\begin{array}{cccccc} 1 & x_1 & x^2_1 ...
10
votes
2answers
130 views

Can you prove My conjecture about Invertiblity of the Derivative Matrix ?! (to use Inverse function Theorem)

In the Analysis2 midterm exam, we had the following problem: Let the equation $a_nx^n+\cdots+a_1x+a_0=0$ has $n$ simple real roots (distinct) $\{\alpha_1,\cdots,\alpha_n\}$. Prove that the above ...
2
votes
2answers
79 views

How do you find the determinant of this $(n-1)\times (n-1)$ matrix?

It's for a proof of Cayley's Formula, I know I'm being dumb and can't see it, how do I find the determinant of this $(n-1)\times (n-1)$ matrix where the diagonal entries are $n-1$ and the off diagonal ...