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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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74 votes
2 answers
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Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Characteristic polynomial of a matrix ...
Marc van Leeuwen's user avatar
42 votes
9 answers
13k views

Determinant of a matrix with diagonal entries $a$ and off-diagonal entries $b$ [duplicate]

I have the following $n\times n$ matrix: $$A=\begin{bmatrix} a & b & \ldots & b\\ b & a & \ldots & b\\ \vdots & \vdots & \ddots & \vdots\\ b & b & \ldots &...
M.B.M.'s user avatar
  • 5,426
825 votes
18 answers
173k views

What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
Jamie Banks's user avatar
  • 13.1k
48 votes
4 answers
19k views

How can we prove Sylvester's determinant identity?

Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$ \det(I_m+AB) = \det(I_n+BA)$$ where $I_m$ and $I_n$ denote the $m \times m$ ...
Bruce George's user avatar
  • 1,970
87 votes
12 answers
218k views

How to show that $\det(AB) =\det(A) \det(B)$?

Given two square matrices $A$ and $B$, how do you show that $$\det(AB) = \det(A) \det(B)$$ where $\det(\cdot)$ is the determinant of the matrix?
Learner's user avatar
  • 2,706
39 votes
7 answers
27k views

Determinant of a block lower triangular matrix

I'm trying to prove the following: Let $A$ be a $k\times k$ matrix, let $D$ have size $n\times n$, and $C$ have size $n\times k$. Then, $$\det\left(\begin{array}{cc} A&0\\ C&D \end{array}\...
Buddy Holly's user avatar
  • 1,189
18 votes
8 answers
11k views

How to calculate the determinant of all-ones matrix minus the identity? [duplicate]

How do I calculate the determinant of the following $n\times n$ matrices $$\begin {bmatrix} 0 & 1 & \ldots & 1 \\ 1 & 0 & \ldots & 1 \\ \vdots & \vdots & \ddots & \...
Mohan's user avatar
  • 15k
158 votes
8 answers
246k views

Show that the determinant of $A$ is equal to the product of its eigenvalues

Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$. So I'm having a tough time figuring this one out. I know that I have to work with the characteristic ...
onimoni's user avatar
  • 6,456
25 votes
5 answers
25k views

How to compute the determinant of a tridiagonal Toeplitz matrix?

How to show that the determinant of the following $(n\times n)$ matrix $$\begin{pmatrix} 5 & 2 & 0 & 0 & 0 & \cdots & 0 \\ 2 & 5 & 2 & 0 & 0 & \cdots & ...
user46450's user avatar
  • 321
31 votes
4 answers
14k views

Determinant of rank-one perturbations of (invertible) matrices

I read something that suggests that if $I$ is the $n$-by-$n$ identity matrix, $v$ is an $n$-dimensional real column vector with $\|v\| = 1$ (standard Euclidean norm), and $t > 0$, then $$\det(I + t ...
Stefan Smith's user avatar
  • 8,202
84 votes
6 answers
41k views

Why is the determinant the volume of a parallelepiped in any dimensions?

For $n = 2$, I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculating the area by coordinates. But how can one easily realize that it is true ...
ahala's user avatar
  • 3,050
88 votes
12 answers
80k views

Why determinant of a 2 by 2 matrix is the area of a parallelogram?

Let $A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}$. How could we show that $ad-bc$ is the area of a parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+b, c+d)$? Are the areas of the ...
user avatar
31 votes
8 answers
32k views

Show that the area of a triangle is given by this determinant

I'm not sure how to solve this problem. Can you guys provide some input/hints? Let $A=(x_1,y_1)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in $\mathbb{R}^{2}$. Show that the area of $\...
uohzxela's user avatar
  • 1,637
112 votes
7 answers
114k views

Geometric interpretation of $\det(A^T) = \det(A)$

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns, could someone give a geometric interpretation of the property?
dfg's user avatar
  • 3,941
33 votes
3 answers
30k views

Vandermonde determinant by induction

For $n$ variables $x_1,\ldots,x_n$, the determinant $$ \det\left((x_i^{j-1})_{i,j=1}^n\right) = \left|\begin{matrix} 1&x_1&x_1^2&\cdots & x_1^{n-1}\\ 1&x_2&x_2^2&...
Alec Teal's user avatar
  • 5,525
15 votes
2 answers
18k views

Is adjoint of singular matrix singular? What would be its rank?

Let $A$ be a square and singular matrix of order $n$. Is $\operatorname{adj}(A)$ necessarily singular? What would be the rank of $\operatorname{adj}(A)$?
user61681's user avatar
  • 420
31 votes
6 answers
50k views

Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
najayaz's user avatar
  • 5,489
27 votes
2 answers
26k views

Characterization of positive definite matrix with principal minors

A symmetric matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, ...
Tara's user avatar
  • 444
23 votes
2 answers
56k views

Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory

In order to prove the following identity: $$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ Instead of checking this by brute force, Landau writes thr product of ...
jinawee's user avatar
  • 2,595
119 votes
2 answers
127k views

Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby ...
goblin GONE's user avatar
39 votes
1 answer
33k views

Geometric meaning of the determinant of a matrix

What is the geometric meaning of the determinant of a matrix? I know that "The determinant of a matrix represents the area of ​​a rectangle." Perhaps this phrase is imprecise, but I would like to know ...
Mark's user avatar
  • 7,878
8 votes
4 answers
3k views

Proof If $AB-I$ Invertible then $BA-I$ invertible. [duplicate]

I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
JaVaPG's user avatar
  • 2,726
7 votes
1 answer
738 views

Calculating the determinant gives $(a^2+b^2+c^2+d^2)^2$?

I need to calculate the following determinant in order to prove the following equality: $$\det\begin{pmatrix} a & b & c & d \\ -b & a & -d & c \\ -c & d & a & -b \\...
Vazrael's user avatar
  • 2,291
38 votes
6 answers
46k views

How to prove $\det \left(e^A\right) = e^{\operatorname{tr}(A)}$?

Prove $$\det \left( e^A \right) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}^{n \times n}$.
John's user avatar
  • 381
3 votes
2 answers
980 views

Calculating determinant with different numbers on diagonal and x everywhere else

I'm having troubles solving the following determinant: $$\left| \begin{array}{cccc} a_1 & x & \ldots & x \\ x & a_2 & \ldots & \vdots \\ \vdots & \ldots & ...
Luke's user avatar
  • 153
34 votes
1 answer
12k views

Proof that determinant rank equals row/column rank

Let $A$ be a $m \times n$ matrix with entries from some field $F$. Define the determinant rank of $A$ to be the largest possible size of a nonzero minor, i.e. the size of the largest invertible square ...
spin's user avatar
  • 12k
31 votes
5 answers
13k views

Coefficients of characteristic polynomial of a matrix

For a given $n \times n$-matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed by the columns and rows with indices from $J$. If the characteristic polynomial of $...
Ahia Cohen's user avatar
19 votes
7 answers
36k views

Row swap changing sign of determinant

I was wondering if someone could help me clarify something regarding the effect of swapping two rows on the sign of the determinant. I know that if $A$ is an $n\times n$ matrix and $B$ is an $n\times ...
user133993's user avatar
18 votes
3 answers
1k views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\,\det\left(\pi^{-1}{A}\right)\right|^{-1/2}=\pi^{n/...
0xbadf00d's user avatar
  • 13.7k
16 votes
3 answers
24k views

Proofs of determinants of block matrices [duplicate]

I know that there are three important results when taking the Determinants of Block matrices $$\begin{align}\det \begin{bmatrix} A & B \\ 0 & D \end{bmatrix} &= \det(A) \cdot \det(D) \ \ ...
Perturbative's user avatar
  • 13.1k
12 votes
1 answer
18k views

Log-Determinant Concavity Proof

Can you please help me understand how he gets the equation marked by red from the above one ?
Mohamed Abdelaal's user avatar
12 votes
5 answers
5k views

How to compute the determinant of this Toeplitz matrix?

Given a positive integer $n$, express$$ f_n(x) = \left|\begin{array}{c c c c c} 1 & x & \cdots & x^{n - 1} & x^n\\ x & 1 & x & \cdots & x^{n - 1} \\ \vdots & x &...
Ѕᴀᴀᴅ's user avatar
  • 34.4k
124 votes
11 answers
535k views

What does it mean to have a determinant equal to zero?

After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$. I hope someone can explain this ...
user2171775's user avatar
  • 1,375
47 votes
8 answers
14k views

Why is the determinant of a symplectic matrix 1?

Suppose $A \in M_{2n}(\mathbb{R})$. and$$J=\begin{pmatrix} 0 & E_n\\ -E_n&0 \end{pmatrix}$$ where $E_n$ represents identity matrix. if $A$ satisfies $$AJA^T=J.$$ How to figure out $$\det(A)=...
Laura's user avatar
  • 4,689
43 votes
4 answers
7k views

Slick proof the determinant is an irreducible polynomial

A polynomial $p$ over a field $k$ is called irreducible if $p=fg$ for polynomials $f,g$ implies $f$ or $g$ are constant. One can consider the determinant of an $n\times n$ matrix to be a polynomial in ...
user31559's user avatar
  • 574
34 votes
1 answer
121k views

Using the Determinant to verify Linear Independence, Span and Basis

Can the determinant (assuming it's non-zero) be used to determine that the vectors given are linearly independent, span the subspace and are a basis of that subspace? (In other words assuming I have a ...
eWizardII's user avatar
  • 773
28 votes
4 answers
23k views

Why is determinant a multilinear function?

I am trying to understand (intuitive explanation will be fine) why determinant is a multilinear function and therefore to learn how elementary row operation affect the determinant. I understand that ...
gbox's user avatar
  • 13k
23 votes
7 answers
20k views

Proving the relation $\det(I + xy^T ) = 1 + x^Ty$

Let $x$ and $y$ denote two length-$n$ column vectors. Prove that $$\det(I + xy^T ) = 1 + x^Ty$$ Is Sylvester's determinant theorem an extension of the problem? Is the approach the same?
Rusty's user avatar
  • 2,888
22 votes
4 answers
97k views

The determinant of adjugate matrix

I have the following proof that I would like to be walked through because I'm not intuitively seeing what to do: If $A$ is $n\times n$, prove $\det\left(\operatorname{adj}(A)\right) = \det(A)^{n-1}$. ...
n8sty's user avatar
  • 513
12 votes
2 answers
7k views

Is sum of two orthogonal matrices singular?

I am trying to solve following problem. Let $A, B \in \mathbb{R}^{n\times n}$ be an orthogonal matrices and $\det(A) = -\det(B)$. How can it be proven that $A+B$ is singular? I could start with ...
chip's user avatar
  • 161
5 votes
1 answer
1k views

Question on determinants of matrices changing between integer matrices [duplicate]

The following problem came up from a though I had while reading: Let's say we have $M=\mathbb{Z}^n$ and we have another free $\mathbb{Z}$-module, $N$, inside of $M$ also with rank $n$. We know we ...
Atticus Christensen's user avatar
3 votes
2 answers
2k views

Matrix determinant lemma with adjugate matrix

I would like a proof of the following result, given on wikipedia. For all square matrices $\mathbf{A}$ and column vectors $\mathbf{u}$ and $\mathbf{v}$ over some field $\mathbb{F}$, $$ \det(\mathbf{A}...
user44090's user avatar
63 votes
1 answer
126k views

Effect of elementary row operations on determinant?

1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ...
dfg's user avatar
  • 3,941
60 votes
3 answers
20k views

What is the origin of the determinant in linear algebra?

We often learn in a standard linear algebra course that a determinant is a number associated with a square matrix. We can define the determinant also by saying that it is the sum of all the possible ...
user8210's user avatar
  • 1,093
53 votes
2 answers
7k views

Development of the Idea of the Determinant

While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better ...
analysisj's user avatar
  • 2,700
24 votes
1 answer
6k views

Invertible matrices over a commutative ring and their determinants

Why is it true that a matrix $A \in \operatorname{Mat}_n(R)$, where $R$ is a commutative ring, is invertible iff its determinant is invertible? Since $\det(A)I_n = A\operatorname{adj}(A) = \...
Tom Oldfield's user avatar
  • 13.1k
19 votes
4 answers
14k views

Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?

This is a statement in linear algebra that I can't seem to understand the proof behind. For a square matrix $A$, why is: $$\mathrm{adj}(A)A = \det(A) \cdot I$$ Any explanation would be greatly ...
vondip's user avatar
  • 1,803
14 votes
7 answers
10k views

$\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$

How can I prove that $\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$, if $A$ and $B$ are any two $n\times n$-matrices. Here, $\operatorname{adj} A$ means the adjugate of the ...
vika_bar's user avatar
  • 149
14 votes
4 answers
3k views

Determinant of a specific circulant matrix, $A_n$

Let $$A_2 = \left[ \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right]$$ $$A_3 = \left[ \begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$ $$...
nullUser's user avatar
  • 28k
12 votes
3 answers
6k views

The determinant function is the only one satisfying the conditions

How can I prove that the determinant function satisfying the following properties is unique: $\det(I)=1$ where $I$ is identity matrix, the function $\det(A)$ is linear in the rows of the matrix and ...
Richard Nash's user avatar

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