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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questions
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The "second derivative test" for $f(x,y)$
I'm currently taking multivariable calculus, and I'm familiar with the second partial derivative test. That is, the formula $D(a, b) = f_{xx}(a,b)f_{yy}(a, b) - (f_{xy}(a, b))^2$ to determine the ...
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votes
2
answers
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Prove that $\det(AB-BA)=0$
Let $A,B$ be two $3 \times 3$ matrices with complex entries such that $$(A-B)^2=O_3$$
Prove that $$\det(AB-BA)=0$$
I tried to prove this with ranks. I denoted $X=A-B$ and thus $X^2=O_3$ which means ...
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votes
8
answers
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Alternative definition of the determinant of a square matrix and its advantages?
Usually, the definition of the determinant of a $n\times n$ matrix $A=(a_{ij})$ is as the following:
$$\det(A):=\sum_{\sigma\in S_n}\text{sgn}(\sigma)\prod_{i=1}^na_{i,\sigma(i)}.$$
In Gilbert ...
user9464
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votes
3
answers
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Prove/disprove: if $\det(A+X) = \det(B + X)$ for all $X$, then $A=B$
I have to prove/disprove this:
If $\det(A+X) = \det(B + X)~ \forall X \in M_{n \times n} (\mathbb F) \rightarrow A = B$
I believe it is true but I can not think of a direct way to prove it.
Any ...
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votes
5
answers
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What is Cramer's rule used for?
Cramer's rule appears in introductory linear algebra courses without comments on its utility. It is a flaw in our system of pedagogy that one learns answers to questions of this kind in courses only ...
14
votes
1
answer
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Sign of determinant when using $det A^\top A$
We have been given matrix:
$$A =
\begin{pmatrix}
a& b& c &d \\
b &−a& d& −c\\
c& −d &−a& b \\
d &c& −b& −a\\
...
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votes
3
answers
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$Tr(A^2)=Tr(A^3)=Tr(A^4)$ then find $Tr(A)$
Let $A$ be a non singular $n\times n$ matrix with all eigenvalues real and
$$Tr(A^2)=Tr(A^3)=Tr(A^4).$$Find $Tr(A)$.
I considered $2\times 2$ matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ ...
user312648
13
votes
2
answers
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Do determinants of binary matrices form a set of consecutive numbers?
While pondering a solution for the problem of generating random 0-1 matrices with small absolute determinants, I once again realise how little I know about 0-1 matrices. My initial idea was to pick a ...
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votes
1
answer
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Determinant game - winning strategy [duplicate]
I came across this problem while looking at Putnam problems a while ago:
Alan and Barbara play a game in which they take turns filling entries of an initially empty $2008 \times 2008$ array. Alan ...
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votes
3
answers
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Determinant of a companion matrix
I have to find determinant of $$A := \begin{bmatrix}0 & 0 & 0 & ... &0 & a_0 \\ -1 & 0 & 0 & ... &0 & a_1\\ 0 & -1 & 0 & ... &0 & a_2 \\ 0 &...
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Let $\left\{\Delta_1,\Delta_2,.....,\Delta_n\right\}$ be the set of all determinants of order 3 that can be made with the distinct real numbers
Let $\left\{\Delta_1,\Delta_2,.....,\Delta_n\right\}$ be the set of all determinants of order 3 that can be made with the distinct real numbers from the set $S=\left\{1,2,3,4,5,6,7,,8,9\right\}$.Then ...
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votes
2
answers
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How many entries in $3\times 3$ matrix with integer entries and determinant equal to $1$ can be even? [duplicate]
Let $A$ be a $3\times 3$ matrix with integer entries such that $\det(A)=1$. At most how many entries of $A$ can be even?
I get a possible solution as $6$ by considering the $3 \times 3$ identity ...
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votes
4
answers
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Determinant of a sum of matrices
I would like to know if the following formula is well known and get some references for it.
I don't know yet how to prove it (and I am working on it), but I am quite sure of its validity, after ...
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votes
2
answers
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Is there a formula for the determinant of the wedge product of two matrices?
I was going over the Wikipedia page for exterior products of vector spaces and we can define the determinant as the coefficient of the exterior product of vectors with respect to the standard basis ...
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votes
2
answers
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Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.
Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.
I was thinking of showing $A$ is not invertible, meaning it has an eigenvalue of $0$. Since no matter ...
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votes
2
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is $\det(A^2 + I)$ always non negative?
Obviously $\det(A^2)$ is (casework), but is the above matrix non-negative? $\det(A)\det(A) \geq 0$ as $\det(A) > 0$ or $\det(A) < 0$ yields positive when squared. However, I am not sure that ...
user624697
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votes
2
answers
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Proof that determinant is continuous using $\epsilon-\delta $ definition
I need to prove that the determinant $\det: M(n, \mathbb{R}) \rightarrow \mathbb{R}$ is a continuous function given the euclidean metric on the vector space of all $n \times n$ matrices over $\mathbb{...
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votes
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answers
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how does addition of identity matrix to a square matrix changes determinant?
Suppose there is $n \times n$ matrix $A$. If we form matrix $B = A+I$ where $I$ is $n \times n$ identity matrix, how does $|B|$ - determinant of $B$ - change compared to $|A|$? And what about the case ...
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I get a wrong determinant - why?
I'm trying to calculate the following determinant:
$$\begin{vmatrix}
a_0 & a_1 & a_2 & \dots & a_n \\
a_0 & x & a_2 & \dots & a_n \\
a_0 & a_1 & x & \dots &...
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votes
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answer
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If $p+q+r=0$, find the value of the determinant
If $p+q+r=0$, prove that the value of the determinant
$$ \Delta= \begin{vmatrix}
pa & qb &rc \\
qc & ra &pb\\
rb& pc & qa \\
\end{vmatrix} =-pqr \begin{vmatrix}
a & b &...
9
votes
1
answer
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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.
...
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votes
1
answer
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Determinant of block tridiagonal Toeplitz matrices
Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case
$$A = \begin{pmatrix} ...
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votes
1
answer
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Can we say that there exist an integer $n$ such $A+nB$ invertible?
If $A$ and $B$ are $3\times 3$ matrices and $A$ is invertible, then can we say that there exists an integer $n$ such that $A+nB$ invertible?
I was trying to show this by choosing $n$ such that ...
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Prove that the Pfaffian satisfies $\text{Pf}(MAM^T)=\det(M)\text{Pf}(A)$
Show that $$\text{Pf} MAM^T = \text{det}M \cdot \text{Pf} A$$ for any matrix $M$ and antisymmetric $A$.
Attempt: $$\text{Pf} MAM^T = \frac{1}{2^N N!} \epsilon_{\alpha_1 \dots \alpha_{2N}} (MAM^T)_{\...
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votes
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answer
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Area of triangle in determinant form
Area of triangle with vertex $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ is given by :
$$\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1\\x_2 & y_2 & 1\\x_3 & y_3 & 1 \end{vmatrix}$$
In this ...
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Show that if $X \succeq Y$, then $\det{(X)}\ge\det{(Y)}$
Assume that two symmetric positive definite matrices $X$ and $Y$ are such
that $X-Y$ is a positive semidefinite matrix. Show that $$\det{(X)}\ge\det{(Y)}$$
I felt this result is clear, but I can't ...
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Suppose A is an n-by-n matrix with its diagonal entries are n and other entries are one. Find determinant of A.
For $n \geq 2$, find the determinant of
$A_{n}=\begin{bmatrix}
n & 1 & 1 &\ldots &1 \\
1 & n & 1 &\ldots &1 \\
1 & 1 & n &\ldots &1 \\
\vdots & \...
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determinant of a matrix with binomial coefficient entries
I trying to prove a statement, which boils down to showing that the determinant of a specific matrix is nonzero. I use the convention that $\binom{n}{k} = 0$
if $k > n$ or $k < 0$. Let $k,l$ be ...
user538454
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answers
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Finding $\det(I+A^{100})$ where $A\in M_3(R)$ and eigenvalues of $A$ are $\{-1,0,1\}$
I have a matrix $A \in M_3(R)$ and it is known that $\sigma (A)=\{-1, 0, 1\}$, where $\sigma (A)$ is a set of eigenvalues of matrix $A$. I am now supposed to calculate $\det(I + A^{100})$.
I know ...
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votes
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answers
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Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$
Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues.
If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be ...
- 12.5k
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votes
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Matrix determinant lemma derivation
While reading this wikipedia article on the determinant lemma, I stumbled upon this expression (in a proof section):
\begin{equation}
\begin{pmatrix} \mathbf{I} & 0 \\ \mathbf{v}^\mathrm{T} & ...
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votes
2
answers
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views
Determinant of a $2 \times 2$ complex block matrix is nonnegative
Let $n \geq 1$ and $A, B \in M_n(\mathbb C)$. Form the matrix
$$g=
\begin{bmatrix}
A & -B \\
\overline B & \overline A
\end{bmatrix}
\in M_{2n}(\mathbb C)$$
I would like to prove ...
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votes
0
answers
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views
Proof of the conjecture that the kernel is of dimension 2, extended
Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
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votes
3
answers
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Easiest way to find characteristic polynomial for this 4x4 matrix
I have been given the matrix
$$
\begin{bmatrix}
1 & 3 & 0 & 3 \\
1 & 1 & 1 & 1 \\
0 & 4 & 2 & 8 \\
2 & 0 & 3 & 1 \\
\end{bmatrix}
$$
and told I must ...
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votes
1
answer
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Winning strategy in $(2n+1) \times (2n+1)$ matrix game.
Edit: A few minutes after posting this question (that I had been thinking about for about a day) I figured out the answer in the $3 \times 3$ case; see my answer below. However, the question might ...
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answers
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To show: $\det\left[\begin{smallmatrix} -bc & b^2+bc & c^2+bc\\ a^2+ac & -ac & c^2+ac \\ a^2+ab & b^2+ab & -ab \end{smallmatrix}\right]=(ab+bc+ca)^3$
I've been having quite some trouble with this question.
I'm required to show that the below equation holds, by using properties of determinants (i.e. not allowed to directly expand the determinant ...
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votes
1
answer
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The average determinant of all integer matrices with coefficients $0,1,2$ [closed]
Let $S$ denote the set of $A \in M(n,\mathbb R)$ such that every entry of $A$ is either of $0$, $1$ or $2$, then is it true that $$\sum_{A \in S} \det A \ge 3^{n^2}\ ?$$
user228168
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answers
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Inverse of a sum of positive definite matrices
Let $A,B$ be symmetric positive definite matrices. Let $A^{-1} = LL^T$ (Cholesky decomposition, $L$ is lower-triangular). I think the following identities are true, but I haven't found them online:
$$
...
user14559
6
votes
1
answer
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Drawing a tetrahedron from a parellelepiped to convince myself it is 1/6th the volume,
I drew a parallelepiped that is spanned by three vectors, and we know the volume is given by the absolute value of the determinant of the matrix - with the three vectors arranged in rows (or columns, ...
- 1
6
votes
1
answer
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views
Filling an array(Putnam)
Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a ...
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votes
1
answer
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Cramer's rule: Geometric Interpretation
I have a question concerning Cramer's rule:
Let $A$ be a matrix and $A \cdot \vec x = \vec b$ a lineare equation. $A_i$ is the matrix $A$ where the i'th column is replaced by $\vec b$
if $det(A) \...
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votes
4
answers
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views
$\det(I+A)=1+\operatorname{Tr}(A)$ if $\operatorname{rank}(A)=1$
Let $A$ be a complex matrix of rank $1$. Show that $$\det (I+A) = 1 + \operatorname{Tr}(A)$$ where $\det(X)$ denotes the determinant of $X$ and $\operatorname{Tr}(X)$ denotes the trace of $X$.
Any ...
user598858
6
votes
2
answers
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views
nth derivative of determinant wrt matrix
I'm working on an expression for the nth derivative of a (symmetric) matrix, i.e.
\begin{equation}\frac{\partial^{n} \det(A)}{\partial A^{n}}\end{equation}
Starting with \begin{equation}\frac{\partial ...
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votes
1
answer
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views
How is the matrix identity $\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B)$ proved?
The Wikipedia page about the determinant mentions the following matrix identity
$$\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B),$$
valid for squared matrices $A$ and $B$ of the ...
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votes
2
answers
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views
If $\vert A\vert+\vert B\vert =0,$ then What is the value of $\vert A+B\vert$?
There are two square matrices $A$ and $B$ of same order such that
$A^2=I$ and $B^2=I,$Where $I$ is a unit matrix.If $\vert A\vert+\vert
B\vert =0,$ then find the value of $\vert A+B\vert ,$here $\...
- 3,539
5
votes
3
answers
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Polynomial form of $\det(A+xB)$
Let $A$ and $B$ be two $2 \times 2$ matrices with integer entries. Prove that $\det(A+xB)$ is an integer polynomial of the form $$P(x) = \det(A+xB) = \det(B)x^2+mx+\det(A).$$
I tried expanding the ...
- 15.6k
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votes
5
answers
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How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$
Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix
$$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$
How to prove that ...
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votes
1
answer
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Characterising minors of diagonal matrices
Let $k,d$ be positive integers, $1<k<d$. Let $\lambda_I=\lambda_{i_1,\ldots,i_k}$ be real numbers, indexed by multi-indices $I=(i_1,\ldots,i_k)$, where $1\le i_1<\ldots<i_k \le d$.
Are ...
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votes
1
answer
886
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Determinant evaluation for matrix with $-1, 2, -1$ below/on/above diagonal [duplicate]
What is the trick for evaluating the determinant of this matrix?
$$\begin{bmatrix}
2 & -1 \\
-1 & 2 & -1 \\
& -1 & 2 & -1 \\
&& -1 & 2 & -1 \\
&&& -...
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5
votes
1
answer
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Find the determinant of the $n\times n$ matrix $A_n$ with $(A_n)_{i,j}={n\choose |i-j|}$.
I'd like to find the determinant of the matrix $A_n$ given by $(A_n)_{i,j}={n\choose |i-j|}$ for all $n\in\mathbb{Z}_{\ge 1}$ and $i,j\in\{1,2,\ldots,n\}$. Here is what I know so far:
$\det(A_n)=0$ ...
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