Questions tagged [determinant]
Questions about determinants: their computation or their 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.
6,892
questions
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Theorem 9.34 Rudin
I don't understand what do we mean in $j_q$ and $j_p$.
I also don't understand part $(b)$ of 9.34 theorem.
I could n't understand this sentence: "det is a linear function of each of the column ...
1
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2
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208
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Use row reductions to show $det(T)=0$
Use row operations to show that
$det(T)=0$,
where
$$T = \begin{bmatrix}
x^2 & 2x+1 & 4x+4 & 6x+9\\
y^2 & 2y+1 & 4y+4 & 6y+9\\
z^2 & 2z+1 & 4z+4 & 6z+9\\
w^2 &...
1
vote
1
answer
217
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Show sets of dual vectors span the same subspace iff wedge products are similar
The original question is as follows:
Let $(f_1, f_2, ..., f_k)$ and $(g_1, g_2, ..., g_k)$ be linearly independent sets of dual vectors in $(\mathbb{R}^n)^*$.
Show that both sets span the same k-...
1
vote
1
answer
64
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Proving that three functions are linearly independent.
Three functions $f_1,\ f_2\ f_3$ with domain $R$ is linearly dependent if there are three numbers $\lambda_1\lambda_2\lambda_3$ that are not all 0 and $\lambda_1f_1(x)+\lambda_2f_2(x)+\lambda_3f_3(x)=...
0
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0
answers
121
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Relationship between rank and determinant
Let $A$ be an $n\times n$ matrix and rank of $A$ be $r$. If we multiply 2 to every row. What will the new determinant be? $2^r |A|$ or $2^n |A|$
I can understand why $2^n |A|$ is the answer but is $2^...
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0
answers
52
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Possible Proof that determinant of submatrix of Unitary matrix is alswys greater 0?
Is there a method to show that $\det(R) > 0$ with $R^{M\times M}$ as a submatrix of an orthogonal matrix $Q^{N\times N}$, with $QQ^T=I_N$, $M<N$.
I would be grateful for a helpful hint.
3
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0
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536
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Proving the two matrices generate the group $SL_2(\mathbb{Z})$ of $2\times 2$ integer matrices with determinant 1.
Can someone please help me apply elementary row and column operations to prove that two matrices $g = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}$ and $h = \begin{pmatrix} 0 & -1 \\ 1 &...
0
votes
1
answer
45
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generalized cross product with proving the perpendicularilty
I understand this is how we define the generalized vector product for higher dimensions. That is for $v_1,...,v_{n-1}$ linearly independent, write $w$ vector with component $w_i=det(v_1,...,v_{n-1},...
2
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0
answers
42
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How to prove this Proposition about determinant functions and linear composition?
Let $\Delta$be a determinant function of $N$ dim vector space $V$.
Let $|v\rangle$and $\{|v_k \rangle \}^N_{k = 1} $ in vector space $V$.
Proof the following eqatuion:
$$\sum_{j=1}^N(-1)^{j-1}\Delta(|...
0
votes
1
answer
257
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Determinant of a quadratic form.
Is there any visual or not, interpretation of the determinant of a quadratic form? I think it helps me to understand why the determinant is not an invariant under a change of basis. Because the ...
0
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1
answer
33
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Show that equations system has only one solution
Show that the equation system has only one solution
$$ax+by+cz=d_1\\cx+ay+bz=d_2\\x+y+z=d_3$$
Where a, b, c are real number that are not all the same, $d_1,\ d_2,\ d_3$ are any real numbers.
I get the ...
6
votes
1
answer
191
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Understanding determinant $=0$
I am playing around with determinants to see if I can get a better grasp of it and would appreciate some thoughts.
$$A=\begin{pmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{...
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38
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On computing the determinant of a certain kind of matrices
I would like to compute the determinant of a matrix of the following type:
\begin{pmatrix}
a_0 & a_1 & a_2 & \dots & a_{n-2} & a_{n-1} \\
a_1 & a_0 & ...
0
votes
1
answer
126
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Find the determinant of matrix B, using matrix A and elementary row operations.
I have matrix A which is
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}
with a determinant of -4; and matrix B which is
\begin{...
4
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1
answer
111
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Show that the equation for a plane can be expressed with determinant
Show that the equation for a plane through the points $(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)$ can be written as
$$\begin{vmatrix}x&y&z&1\\x_1&y_1&z_1&1\\x_2&y_2&z_2&...
1
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0
answers
35
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Find coefficients of a functional determinant.
Let us consider the functional determinant
\begin{gather*}
\Delta=\begin{vmatrix} P_{11}(x,y) & P_{12}(x,y) &P_{13}(x,y)\\
P_{21}(x,y) & P_{22}(x,y) &P_{23}(x,y)\\
P_{31}(x,y) & ...
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1
answer
42
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Can I infer linear dependence from the following span relations?
Consider a $d$-dimensional complex vector space. I have three sets of $d-1$ vectors each for which the following holds:
$\text{span}(v_1,...,v_{d-2},v_{\alpha})=\text{span}(v_1,...,v_{d-2},v_{\beta})=\...
2
votes
1
answer
63
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Determinant of a matrix with non-square properties
Let A be an n x n matrix such that the det(A)=5;
Let E be an m x m matrix such that the det(E)=4;
Let F be an n x m matrix.
Find the det\begin{bmatrix}0&A\\E&F\end{bmatrix}
The answer can be ...
0
votes
1
answer
319
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How determinant = 0 proves that the equation system has either none or infinite solutions.
I'm trying to grasp the validity of $det(A)=0$ means the equations system , $A\vec{x}={y}$, has either infinite solutions or none, depending on $\vec{y}$.
In two or three dimensions the graphical ...
2
votes
1
answer
80
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the set of matrices $A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ satisfies following properties
consider a matrix $A\in\mathbb{R^{2\times2}}$
$$A=\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$$
and let $A$ has the following properties
$\bullet$ $A$ is not invertible
$\bullet$ $A$ is ...
1
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0
answers
131
views
Prove that $det$ and $tr$ are continuous.
Exercise 6 after $\S$ 91 from Paul R. Halmos's Finite-Dimensional Vector Spaces (second edition) invites to prove or disprove the following assertion.
Prove that $det$ and $tr$ are continuous.
I ...
3
votes
1
answer
87
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Determinant of a $4 \times 4$ matrix
If $$ A = \begin{pmatrix}a&b&c&d\\-b&a&-d&c\\-c&d&a&-b\\-d&-c&b&a\end{pmatrix} $$ calculate $\det(A)$.
If you calculate
$$AA^t=\begin{pmatrix}a^2+b^2+c^...
8
votes
1
answer
307
views
How to prove the following determinant identity
Prove:
$$
\begin{array}{|cccccccccc|} 1 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & \cdots & 0 \\ x & x & x & \cdots & x & y & y & y & \cdots ...
0
votes
1
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138
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Linear algebra: nilpotent matrix and determinant.
My linear book had an exercise that demonstrated that a nilpotent matrix A has det(A)=0
$A^k=0$ is the nilpotent condition.
$det(A^k)=(det(A))^k$ and since $det(A^k)=0 \Leftrightarrow det(A)=0$.
My ...
1
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1
answer
756
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Determinant of cyclic matrix, proof without eigenvectors.
I tried, in vain, to prove the following formula for the determinant of a cyclic matrix:
$$
\begin{vmatrix}a_1&a_2&a_3&\cdots&a_n\\a_n&a_1&a_2&\cdots&a_{n-1}\\a_{n-1}&...
3
votes
0
answers
76
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what's wrong with this matrix (to find the determinant using Laplace expansion)?
I have to compute the determinant of this 4x4 matrix:
\begin{bmatrix}2&1&3&0\\-1&0&1&2\\2&0&-1&-1\\-3&1&0&1\end{bmatrix}
this is what I did:
I swapped ...
0
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0
answers
42
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triple cross product (for beginner)
I was reading triple vector product. I saw an expression like this :
$${\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{x}&=\mathbf {u} _{y}(\mathbf {v} _{x}\...
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0
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Is it sufficient enough to only compose a equation of nonzero value of determinant of a matrix to prove a matrix being invertible?
$$A:=n\times n~\text{matrix}$$
As this matrix is invertible, at least, the following are true.
$$
\begin{equation*}%uasge:&smth\\..Dont write symbol of line break at the end of row
\begin{cases}
...
1
vote
2
answers
294
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Why is the covolume of a lattice is unique?
Given a basis $\mathfrak{B}=\{b_1, b_2, \cdots, b_n\}$ of $\mathbb{R}^n,$ the lattice generated by $\mathfrak{B}$ is the set of all linear combinations with integer coefficients: $$m_1b_1+m_2b_2+\...
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2
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164
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Plane parallel to intersection line using determinants
Determine a so that the intersection line between the planes
$P_1: 2x+ay-z=3$
$P_2: x-2y+az=5$
are parallel to the plane $P_3: 2x+y+z=2$.
I want to solve this using determinants in some way.
Im ...
0
votes
1
answer
33
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Find the determinant of dot product entries matrix under $O(k)$ action
Notation : Here $O(k)$ denotes the orthogonal group, $V_k(\mathbb{R}^n)$ the Stiefel manifold of $k$ orthonormal-frames and define $\sigma : V_k(\mathbb{R}^n) \times V_k(\mathbb{R}^n) \longmapsto \...
9
votes
2
answers
225
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Determinant of matrix with $2$'s and the pattern $3\ 1\ 3$
Let for $n\ge 1, D_n$ be the determinant of the $n\times n$ matrix $A_n$ where the entries along the main diagonal (i.e. of the form $(i,i)$ for $1\leq i\leq n$) are all $3$, entries of the form $(i,i+...
0
votes
0
answers
93
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The knack to reduce the amount of calculations for the value of the determinant of $~5\times 5~$ matrix
$$\det\left(A\right):=
\begin{vmatrix}
3&4&6&3&-1\\
2&1&-3&2&2\\
-2&3&2&1&-3\\
1&1&1&3&-1\\
2&-1&5&-3&6\\
\end{vmatrix}=...
1
vote
1
answer
143
views
Proving $\det(I + uv^t) = 1 + v^t u$ using alternating multilinear map properties of the determinant
Currently, I am reading Linear Algebra Done Right by Sheldon Axler. On page 318, I came across a proof of $\det(A\cdot B) = \det(A)\cdot \det(B)$ that I found quite elegant. This proof utilizes the ...
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0
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Definition of the power of the determinant of a matrix
Suppose, I have a square matrix $A$. Now, what is the definition of $[det(A)]^{n}$? Have mathematicians attempted to define it, or is it such a no-brainer/straightforward that they never attempted to ...
2
votes
1
answer
74
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Does this geometric characterisation of the determinant lead to the usual formal one (multilinear, alternating, unique)
Let the $c_i$ be column vectors of a matrix (rows could equivalently be used).
The formal definition of the determinant (that I'm familiar with) is as follows:
$\det(c_1,c_2,\cdots,c_n):\Bbb R^{n\...
0
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1
answer
184
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Using the Vandermonde Determinant to calculate the Jacobian of a transformation
Let $n\geq 2$ be a natural number. Consider the transformation $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ defined by
$$f(x_{1},...,x_{n})=\left ( \sum_{i=1}^{n}x_{i},\sum_{i=1}^{n}x_{i}^{2},...,\sum_{...
1
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1
answer
102
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Derivative of the determinant function on $2\times2$ matrix
I was asked to show the determinant function on $2\times2$ matrix, regarded as
$f:\mathbf{R^4}\rightarrow\mathbf{R}, f(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3$, is differentiable and to find its derivative. ...
0
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0
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32
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Signs of minors of a real matrix with negative diagonal entries, positive off-diagonal entries, and negative row sums
Let $A$ be a real $n\times n$ matrix where the diagonal entries are negative, the off-diagonal entries are positive, the row sums are negative, and sign$(|A|)=(-1)^n$. Let $M$ be the $i,j$ minor of $...
1
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1
answer
48
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What values can the given determinant not take under the given conditions?
Let A & B are two $n\space \times \space n$ matrices with real entries and $|B|\ne0$ then:
$|A^2+I_n|$ can not be?
(a) -1
(b) 1
(c) 2
(d) 0
|$I_n-AB|-|I_n-BA|$ is equal to?
(a) 0
(b) 1
(c) 2
(d) ...
-2
votes
1
answer
47
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Can $x$ take every value here? [closed]
To find $x$ such that this determinant is zero, a,b,c are distinct and real constants.
$$
\begin{vmatrix}
x-a & a^2 & a^3 \\
x-b & b^2 & b^3 \\
x-c & c^2 & c^3 \...
1
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0
answers
57
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Showing $ ( T(p) )(x) = p'(x) + p(0) \cdot x^n $ is diagonalizable
Problem: Let $ V = \mathbb{C_n}[x] $ be the polynomial space of degree at most $n $. We define a linear transformation $ T: V \to V $ as $ ( T(p) )(x) = p'(x) + p(0) \cdot x^n $.
Prove this ...
2
votes
0
answers
37
views
The mathematical structure of tensor invariants
I am studying tensor algebra and came across tensor invariants, and how they are composed of its eigenvalues. What is known is that these invariants are the coefficients of the characteristic ...
1
vote
0
answers
118
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Determinant of Vandermonde matrix [duplicate]
I have the following problem:
Let $(a_{1},\dots,a_{n})\in\mathbb{R}^{n}$ a vector and let the matrix
(Vandermonde) \begin{equation*}
V =
\begin{pmatrix}
1&a_{1}&...
0
votes
1
answer
71
views
The question is from the chapter matrix.
The question was asked in my class of matrix and determinants.
Suppose for two matrices A and B, we have trace(AB) = trace(A)trace(B). Then trace(A²-B²) = ?
a)[trace(AB)]²
b)trace(AB²A)
c)trace(AB)²
d)...
2
votes
1
answer
96
views
Big Matrix Small Determinant
From a programming competition:
Construct a square matrix with $N$ rows and $N$ columns consisting of non-negative integers from $0$ to $10^{18}$, such that its determinant is equal to $1$, and there ...
0
votes
1
answer
171
views
Determinant and symmetric function of eigenvalues
Let $H$ be an $n$-dimensional vector space and $T$ a linear operator on it with eigenvalues $\lambda_{i_1},\dots,\lambda_{i_n}$. Let $I$ be the identity operator and $z\in\mathbb{C}$. How does one ...
0
votes
2
answers
309
views
Volume preserving linear map
I know a result that volume preserving linear maps have determinant 1. How do I prove it? I understand it has to do with change of variables but does “derivative of a linear map” make sense? (...
1
vote
1
answer
83
views
Determinant of $A^T D A$
Let $A$ be an $n \times m$ tall matrix ($n > m$) and let $D$ be a diagonal $n \times n$ matrix. Is the following correct?
$$\det\left( A^T D A \right) = \det \left( A^T A \right) \det(D)$$
-1
votes
2
answers
87
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Matrices Determinant
Let $$ A=
\begin{bmatrix}
-6 & -8 & 15 & 0 & 9 \\
2 & 5 & 4 & 0 & 8 \\
-8 & 7 & -6 & 9 & 1 \\
16&6&-22&8&-20\\
...