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.
1
vote
2answers
40 views
simpler way to calculate a determinant?
Simpler way to calculate this?
$$A = \begin{bmatrix}\lambda -2 & 2 & 0 \\ 2 & \lambda -1 & 2 \\ 0 & 2 & \lambda \end{bmatrix}$$
my method:
\begin{align*} \det A
&= \det ...
0
votes
1answer
30 views
Step in Euler's rotation theorem
I have been examining the matrix proof for Euler's rotation theorem on Wikipedia. I have deduced every step up to proving that $\det (R - I) = 0$ for any rotation matrix R. However, I'm having ...
0
votes
1answer
61 views
Calculate the determinant? any hints
Calculate the determinant
$$\begin{vmatrix}
n & s_1 & s_2 & \cdots & s_{n-1} \\
s_1 & s_2 & s_3 & \cdots & s_n \\
s_2 & s_3 & s_4 & \cdots & s_{n+1} \\
...
1
vote
4answers
117 views
Calculate the Determinant?
$$D=\begin{bmatrix}
246 & 427 & 327 \\
1014 & 543 & 443 \\
-342 & 721 & 621 \\
\end{bmatrix}$$
What's the trick?
Hints?
Of course I know calculate by definition...
Please ...
0
votes
0answers
18 views
Prove that the determinant of a tensor is Invariant in any Basis
Considering the following:
Let be Q the tensor that transform the ON basis ($e_1$, $e_2$, ... $e_n$) in the ON basis ($e'_1, e'_2$, ... $e'_n$).
e'$_j$ = Q $\cdot$ e$_j$
hence we get
e'$_j$ = ...
-3
votes
2answers
30 views
4 Small questions about matrices, eigenvalue, consistence, lineare independant, eigenvector
i've some questions
A)
"A" is a real 5 x 3 matrix y € R³ and z € R^5 and for this Ay = z
Why can you consider that Ax = 4z, is consistent?
B)
...
7
votes
2answers
86 views
Historical meaning and usage of determinant
Can anyone please explain how, why, and where determinants were developed/formalized? What was their historical usage? Why were they initially formulated and what were they used for (and later ...
3
votes
2answers
126 views
Find matrices $X$ such that for any matrix $Y$ we have $\det(X^2 + Y^2) \geq 0$ [duplicate]
What is the characterization of real matrices $X \in \mathbb{R}^{n\times n}$ such that for any real matrix $Y \in \mathbb{R}^{n\times n}$: $$\det(X^2 + Y^2) \geq 0?$$
4
votes
3answers
61 views
A basic question on determinant and rank of a matrix
How to prove that if the determinant of a $n \times n$ matrix is zero then the rank is less than $n$. I can prove the converse. Only a hint is enough.
My definition of rank is the maximum number of ...
0
votes
1answer
22 views
Determinant of elementary matrix of type 2
I would like to have an induction proof of why the determinant of an elementary matrix with two rows swapped equals -1. I'm brand new to determinants. Thanks.
2
votes
2answers
19 views
Determinant of permutation matrix (elementary matrix of type 2)
I would like to know why the determinant of a permutation matrix of size nxn (elementary matrix of size nxn of type 2) is -1. I'm brand new to determinants and I've tried expanding it and using ...
1
vote
0answers
52 views
How to determine that a certain eigenvalue is doubly degenerate?
Given a symmetric matrix $X$. I ask myself, how to determine that a certain eigenvalue $\lambda$ is (exactly) doubly degenerate?
I thought about several approaches:
Calculate the derivative of $ ...
2
votes
2answers
67 views
Calculate the determinant of a matrix multiplied by itself confirmation
If $ \det B = 4$ is then is $ \det(B^{10}) = 4^{10}$?
Does that also mean that $\det(B^{-2}) = \frac{1}{\det(B)^2} $
Or do I have this completely wrong?
7
votes
5answers
196 views
Determine the value of a second determinant based on the first
I know the theory of determinants, but I have no idea how to apply it to this problem.
Suppose $$\det\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i \end{bmatrix} = 6$$
What is the value ...
2
votes
2answers
67 views
Prove that invertible metrices set is an open set in a given space, and the determinant is continuous [duplicate]
Given a matrix $M_{n\times m}$, we can think about it as a vector in $\mathbb{R}^{n\times m}$ (How come?).
How can I prove that the set of all the invertible metrices of size $n\times n$ is an open ...
0
votes
0answers
17 views
Minor, cofactor of a matrix.
(Minor, Cofactor of a Matrix): The number $ \det \left(A(i\vert j)\right)$ is called the $ (i,j)^{\mbox{th}}$ minor of $ A$ . We write $ A_{ij} = \det \left(A(i\vert j)\right).$ The $ ...
4
votes
1answer
69 views
Calculate the determinant
Calculate the determinant
$$\begin{align*}D[n]=\begin{array}{cccccc} b & b & b & \dots & b & a \\ b & b & b & \dots & a & b \\ \vdots & \vdots & ...
0
votes
3answers
90 views
Solving for unknown value using properties of determinant
Problem : If $ax^4 +bx^3+cx^2+dx+e= $ $$
\begin{vmatrix}
x^3+3x & x-1 & x+3 \\
x+1 & -2x & x-4 \\
x-3 & x+4 & 3x \\
\end{vmatrix}
$$
...
1
vote
4answers
66 views
Calculate the determinant of the matrix $(a_{ij})$ where $a_{ij}=a+b$ when $i=j$, and $a_{ij}=a$ otherwise
The matrix is $n\times n$ , defined as the following:
$$
a_{ij}=\begin{cases}
a+b & \text{ when } i=j,\\
a & \text{ when } i \ne j
\end{cases}.
$$
When I calculated it I got the ...
1
vote
1answer
45 views
Prove that the determinant of polynomials is zero
Prove that this determinant is zero (this matrix is $n\times n$):
$$\begin{vmatrix}
f_1(a_1) & f_1(a_2) & \cdots & f_1(a_n) \\
f_2(a_1) & f_2(a_2) & \cdots & f_2(a_n) \\
\vdots ...
1
vote
1answer
45 views
On the difference of two positive semi-definite matrices
I am relatively new to linear algebra, and have been struggling with a problem for a few days now. Say we have two positive semi-definite matrices $A$ and $B$, and further assume that $A$ and $B$ are ...
2
votes
3answers
149 views
Determinant (A + B) = det(A) + det(B)?
Well considering two $n \times n$ matrices does the following hold true:
$$\det(A+B) = \det(A) + \det(B)$$
Can there be said anything about $\det(A+B)$?
If $A/B$ are symmetric (or maybe even of the ...
2
votes
8answers
362 views
Do matrices have a “to the power of” operator?
Well I was sure that saying "$A^3$" (where $A$ is an $n\times n$ matrix) is nonsense. Sure one could do $(A\cdot A) A$ But that contains different operators etc. So what did my prof mean by the ...
1
vote
2answers
56 views
Extension of Sylvester's Theorem
$\mathbf{h}_i\in\mathbb{C}^{M}$ are column vectors $\forall i=\{1, 2, \cdots, K\}$.
$q_i\in\mathbb{R}_+$ are scalars $\forall i=\{1, 2, \cdots, K\}$
$\lvert\bullet\rvert$ denotes determinant of a ...
2
votes
2answers
55 views
Intuition/Understanding of Inverse and Determinants
This is not homework, but extends from a proof in my book.
EDIT
We're given an $m \times m$ nonsingular matrix $B$.
According to the definition of an inverse, we can calculate each element of a ...
1
vote
0answers
40 views
Closed Form for Continuant (Determinant Tridiagonal Matrix)
Consider the particular tridiagonal, $n \times n$ matrix $A$:
A = $\left(\begin{array}{ccccccc}
a_1&b_2 ...
5
votes
0answers
98 views
Linearity of the determinant
I'd like to prove the following properties of the determinant map.
$\det I = 1$
$\det$ is linear in the rows of the input matrix
The determinant map is defined on $n\times n$ matrices $A$ by:
...
3
votes
2answers
44 views
Factorization of a linear combination of matrices
I'm trying to understand the determinant from Axler Sheldon's paper and there is one point in the very beginning that I don't understand :S (Link below to the paper)
...
5
votes
4answers
168 views
$\det(I+A) = 1 + tr(A) + \det(A)$ for $n=2$ and for $n>2$?
Let $I$ the identity matrix and $A$ another general square matrix. In the case $n=2$ one can easily verifies that
\begin{equation}
\det(I+A) = 1 + tr(A) + \det(A)
\end{equation}
or
\begin{equation}
...
1
vote
1answer
72 views
looking for paper by Chapman on determinant of sum of matrices
Math people:
I am trying to find a paper by Chapman referenced after Remark 2.3 in the paper
"On the elementary symmetric functions of a sum of matrices" by
R. S. Costas-Santos posted on the arXiv
...
1
vote
1answer
41 views
Determinant of PSD matrix and PSD submatrix inequality
I'm reading this paper and in the appendix I see the following statement:
For $A \in R^{m\times m}, B \in R^{n\times m}, C \in R^{n\times n}$,
if $D = \begin{bmatrix}A & B\\B^T & ...
2
votes
2answers
63 views
Need help calculating this determinant using induction
This is the determinant of a matrix of ($n \times n$) that needs to be calculated:
\begin{pmatrix}
3 &2 &0 &0 &\cdots &0 &0 &0 &0\\
1 &3 &2 &0 &\cdots ...
5
votes
1answer
43 views
possible determinants of permutations
this is taken from Gilbert Strang's Linear Algebra book:
What are all the possible $4\times4$ determinants of $I + P_{even}$? (P - permutation matrix)
I seem to be stuck on this question except for ...
3
votes
2answers
55 views
Matrix determinant using Laplace method
I have the following matrix of order four for which I have calculated the determinant using Laplace's method.
$$
\begin{bmatrix}
2 & 1 & 3 & 1 \\
4 & 3 & 1 & 4 \\
-1 ...
0
votes
2answers
42 views
Given $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity $a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?
Given positive reals $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity
$a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?
2
votes
0answers
35 views
Determinant with Levi-Civita Symbol
From Schaum's Outline in Tensor Calculus
If $A = [a_{ij}]_{nn} $ is any square matrix, then define $\text{ det } A = \epsilon_{i_1i_2i_3...i_{n-1}i_n}a_{1 \, \cdot \, i_1}a_{2 \, \cdot \, ...
2
votes
2answers
95 views
Elegant proof for nonsingular upper triangular matrix has an upper triangular inverse
I am looking for (an elegant proof or a proof that does not use so many results) that a nonsingular upper triangular matrix $A$ has an upper triangular inverse. Here is what I have:
Nonsingular $\iff ...
1
vote
1answer
47 views
Bounding above the rank of a matrix
I heard about a theorem that states $\text{rank}(X) \leq r$ if the determinant of every $r \times r$ minor of $X$ is zero. Does anyone know of a reference for this theorem or a proof of it?
2
votes
1answer
84 views
Find the determinant of the following general matrix
Let $A_r$ and $B_r$ be the $r\times r$ matrix blocks
$A_r=\left(
\begin{array}{A}
1-t & t^2 & 0 & 0 & 0 & \cdots & 0 \\
t^2 & 0 & 0 & 0 & 0 & \cdots ...
1
vote
2answers
94 views
Determinant of a Matrix Proof: $\;\det(qA) = q^n(\det A)$ [duplicate]
I am required to show that:
$\det(qA) = q^n(\det A)$, where $A$ is a real $n\times n$ Matrix, and $q$ is a constant
I believe that this claim is true after doing few examples. However, but I do not ...
0
votes
1answer
21 views
Determinant of Schur Complement
If I have an $n \times n$ real-valued non-symmetric matrix $\mathbf{M}$, which has determinant $|\mathbf{M}| > 0$, what can I say about the determinant of the matrix $\mathbf{Q}^T \mathbf{M}^{-1} ...
3
votes
4answers
148 views
Prove $\det(kA)=k^n\det A$
Let $A$ be a $n \times n$ invertible matrix, prove $\det(kA)=k^n\det A$. I really don't know where to start. Can someone give me a hint for this proof?
0
votes
2answers
43 views
Linear algebra matrix problem
I was unable to prove the following statement.
Let $A,B$ are both matrices of $n\times n$ of real numbers. We know that the matrix $ A$ is invertible.
Then, we have to prove that there are $n$ ...
-1
votes
1answer
51 views
Find the Wronskian of the Functions [closed]
Find the Wronskian of the functions $f(t)=6e^t\sin{t}$ and $g(t)=e^t\cos(t)$. Simplify your answer.
please list out all steps as simple as possible
thank you
0
votes
1answer
47 views
I solved the question. But I am asking a little bit. $\det(D(fog)(a))=?$
After here, how can I show its determinant?
17
votes
2answers
538 views
Is it always true that $\det(A^2+B^2)\geq0$?
Let $A$ and $B$ be real square matrices of the same size. Is it true that
$$\det(A^2+B^2)\geq0\,?$$
If $AB=BA$ then the answer is positive:
...
2
votes
2answers
104 views
Is determinant uniformly continuous?
The determinant map $\det$ sending an $n\times n$ real matrix to its determiant is continuous since it's a polynomial in the coefficients. Is it also uniformly continuous?
0
votes
0answers
43 views
Pfaffian And Determinant
I am working in tilings using Pfaffian. There is a basic property namely:
Let $ B$ be a $n\times n$ Matrix and
$$ A = \begin{pmatrix}
0 & B\\
-B^T & 0
\end{pmatrix}$$
then
...
0
votes
0answers
38 views
If $D=0$, then what are the necessary conditions for a set of linear equations in three variables to have a unique solution
I have three equations of the form:
a1* x+b1* y+c1* z+d1=0
a2* x+b2* y+c2* z+d2=0
a3* x+b3* y+c3* z+d3=0
what are the necessary conditions for them to have a unique solution if D=0.
I am ...
0
votes
2answers
38 views
The rank of a matrix using the Gaussian method
I need some hints here:
$1.$ By using the Gaussian method how can I calculate the rank of the following matrix.
$$A=\begin{pmatrix} 1&2&\beta\\ 0&\alpha&1\\ 1&0&2\\ ...
