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.
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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 ...
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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 &...
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1
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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-...
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1
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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)=...
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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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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.
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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 &...
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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},...
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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(|...
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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 ...
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1
answer
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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 ...
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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.
$$A=\begin{pmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{...
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On computing the determinant of a certain kind of matrices
I would like to compute the determinant of a matrix of the following type:
\begin{pmatrix}
a_0 & a_1 & a_2 & \dots & a_{n-2} & a_{n-1} \\
a_1 & a_0 & ...
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Find the determinant of matrix B, using matrix A and elementary row operations.
I have matrix A which is
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}
with a determinant of -4; and matrix B which is
\begin{...
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Show that the equation for a plane can be expressed with determinant
Show that the equation for a plane through the points $(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)$ can be written as
$$\begin{vmatrix}x&y&z&1\\x_1&y_1&z_1&1\\x_2&y_2&z_2&...
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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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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})=\...
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$AA^t=\begin{pmatrix}a^2+b^2+c^...
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How to prove the following determinant identity
Prove:
$$
\begin{array}{|cccccccccc|} 1 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & \cdots & 0 \\ x & x & x & \cdots & x & y & y & y & \cdots ...
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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 ...
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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:
$$
\begin{vmatrix}a_1&a_2&a_3&\cdots&a_n\\a_n&a_1&a_2&\cdots&a_{n-1}\\a_{n-1}&...
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what's wrong with this matrix (to find the determinant using Laplace expansion)?
I have to compute the determinant of this 4x4 matrix:
\begin{bmatrix}2&1&3&0\\-1&0&1&2\\2&0&-1&-1\\-3&1&0&1\end{bmatrix}
this is what I did:
I swapped ...
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triple cross product (for beginner)
I was reading triple vector product. I saw an expression like this :
$${\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{x}&=\mathbf {u} _{y}(\mathbf {v} _{x}\...
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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.
$$
\begin{equation*}%uasge:&smth\\..Dont write symbol of line break at the end of row
\begin{cases}
...
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Why is the covolume of a lattice is unique?
Given a basis $\mathfrak{B}=\{b_1, b_2, \cdots, b_n\}$ of $\mathbb{R}^n,$ the lattice generated by $\mathfrak{B}$ is the set of all linear combinations with integer coefficients: $$m_1b_1+m_2b_2+\...
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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 ...
user962204
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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 \...
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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+...
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The knack to reduce the amount of calculations for the value of the determinant of $~5\times 5~$ matrix
$$\det\left(A\right):=
\begin{vmatrix}
3&4&6&3&-1\\
2&1&-3&2&2\\
-2&3&2&1&-3\\
1&1&1&3&-1\\
2&-1&5&-3&6\\
\end{vmatrix}=...
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Proving $\det(I + uv^t) = 1 + v^t u$ using alternating multilinear map properties of the determinant
Currently, I am reading Linear Algebra Done Right by Sheldon Axler. On page 318, I came across a proof of $\det(A\cdot B) = \det(A)\cdot \det(B)$ that I found quite elegant. This proof utilizes the ...
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Definition of the power of the determinant of a matrix
Suppose, I have a square matrix $A$. Now, what is the definition of $[det(A)]^{n}$? Have mathematicians attempted to define it, or is it such a no-brainer/straightforward that they never attempted to ...
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Does this geometric characterisation of the determinant lead to the usual formal one (multilinear, alternating, unique)
Let the $c_i$ be column vectors of a matrix (rows could equivalently be used).
The formal definition of the determinant (that I'm familiar with) is as follows:
$\det(c_1,c_2,\cdots,c_n):\Bbb R^{n\...
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Using the Vandermonde Determinant to calculate the Jacobian of a transformation
Let $n\geq 2$ be a natural number. Consider the transformation $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ defined by
$$f(x_{1},...,x_{n})=\left ( \sum_{i=1}^{n}x_{i},\sum_{i=1}^{n}x_{i}^{2},...,\sum_{...
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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. ...
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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 $...
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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) ...
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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 \...
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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 ...
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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 ...
Stan
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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}&...
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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
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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 ...
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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 ...
user185346
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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? (...
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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)$$
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Matrices Determinant
Let $$ A=
\begin{bmatrix}
-6 & -8 & 15 & 0 & 9 \\
2 & 5 & 4 & 0 & 8 \\
-8 & 7 & -6 & 9 & 1 \\
16&6&-22&8&-20\\
...
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