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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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822 votes
18 answers
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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
280 votes
3 answers
16k views

How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Let $\,A,B,C\in M_{n}(\mathbb C)\,$ be Hermitian and positive definite matrices such that $A+B+C=I_{n}$, where $I_{n}$ is the identity matrix. Show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n \...
math110's user avatar
  • 93.4k
159 votes
8 answers
245k 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
  • 6,466
124 votes
11 answers
534k 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
119 votes
2 answers
126k 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
  • 67.9k
112 votes
7 answers
113k 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
  • 3,921
93 votes
11 answers
8k views

Why does Friedberg say that the role of the determinant is less central than in former times?

I am taking a proof-based introductory course to Linear Algebra as an undergrad student of Mathematics and Computer Science. The author of my textbook (Friedberg's Linear Algebra, 4th Edition) says in ...
dacabdi's user avatar
  • 1,228
91 votes
3 answers
4k views

Cute Determinant Question

I stumbled across the following problem and found it cute. Problem: We are given that $19$ divides $23028$, $31882$, $86469$, $6327$, and $61902$. Show that $19$ divides the following determinant: ...
Potato's user avatar
  • 40.3k
88 votes
12 answers
79k 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
88 votes
6 answers
6k views

Alice and Bob play the determinant game

Alice and Bob play the following game with an $n \times n$ matrix, where $n$ is odd. Alice fills in one of the entries of the matrix with a real number, then Bob, then Alice and so forth until the ...
pad's user avatar
  • 3,017
87 votes
12 answers
217k 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
83 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
  • 3,030
74 votes
6 answers
125k views

Expressing the determinant of a sum of two matrices?

Can $\det(A + B)$ expressed in terms of $\det(A), \det(B), n$ where $A,B$ are $n\times n$ matrices? I made the edit to allow $n$ to be factored in.
Sidharth Ghoshal's user avatar
74 votes
2 answers
12k views

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
71 votes
1 answer
4k views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
Vincent's user avatar
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70 votes
7 answers
102k views

The relation between trace and determinant of a matrix

Let $M$ be a symmetric $n \times n$ matrix. Is there any equality or inequality that relates the trace and determinant of $M$?
TPArrow's user avatar
  • 946
62 votes
1 answer
125k 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
  • 3,921
62 votes
0 answers
2k views

Determinant of a matrix that contains the first $n^2$ primes.

Let $n$ be an integer and $p_1,\ldots,p_{n^2}$ be the first prime numbers. Writing them down in a matrix $$ \left(\begin{matrix} p_1 & p_2 & \cdots & p_n \\ p_{n+1} & p_{n+2} & \...
Rofl Ukulus's user avatar
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
57 votes
3 answers
2k views

What's the sign of $\det\left(\sqrt{i^2+j^2}\right)_{1\le i,j\le n}$?

Suppose $A=(a_{ij})$ is a $n×n$ matrix by $a_{ij}=\sqrt{i^2+j^2}$. I have tried to check its sign by matlab. l find that the determinant is positive when n is odd and negative when n is even. How to ...
Jimmy's user avatar
  • 587
56 votes
10 answers
113k views

Is a matrix $A$ with an eigenvalue of $0$ invertible?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof Suppose $A$ is square matrix and invertible and, for the sake of ...
Derrick J.'s user avatar
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
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
47 votes
5 answers
5k views

Why is the determinant defined in terms of permutations?

Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of ...
Pekov's user avatar
  • 1,045
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
44 votes
5 answers
3k views

Basis-free, field-independent definition of determinants?

Let $T$ be a linear operator on a finite-dimensional vector space $V$ over the field $K$, with $\dim V=n$. Is there a definition of the determinant of $T$ that (1) does not make reference to a ...
WillG's user avatar
  • 6,611
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
43 votes
3 answers
7k views

Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = \...
Mike F's user avatar
  • 22.3k
42 votes
2 answers
20k views

Series expansion of the determinant for a matrix near the identity

The problem is to find the second order term in the series expansion of the expression $\mathrm{det}( I + \epsilon A)$ as a power series in $\epsilon$ for a diagonalizable matrix $A$. Formally, we ...
Spencer's user avatar
  • 12.3k
41 votes
8 answers
18k views

How does Cramer's rule work?

I know Cramer's rule works for 3 linear equations. I know all steps to get solutions. But I don't know why (how) Cramer's rule gives us solutions? Why do we get $x=\frac{\Delta_1}\Delta$ and $y$ and ...
Fawad's user avatar
  • 2,034
41 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
  • 5,406
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
39 votes
5 answers
4k views

Is this determinant always non-negative?

For any $(a_1,a_2,\cdots,a_n)\in\mathbb{R}^n$, a matrix $A$ is defined by $$A_{ij}=\frac1{1+|a_i-a_j|}$$ Is $\det(A)$ always non-negative? I did some numerical test and it seems to be true, but I have ...
Niko's user avatar
  • 771
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
  • 7,866
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
38 votes
3 answers
36k views

Use of determinants

I have been teaching myself maths (primarily calculus) throughout this and last year, and was stumped with the use of determinants. In the math textbooks I have, they simply show how to compute a ...
Argon's user avatar
  • 25.4k
37 votes
3 answers
2k views

How to find the determinant of this $3 \times 3$ Hankel matrix?

Today, at my linear algebra exam, there was this question that I couldn't solve. Prove that $$\det \begin{bmatrix} n^{2} & (n+1)^{2} &(n+2)^{2} \\ (n+1)^{2} &(n+2)^{2} & (n+3)^{2}\\...
Shevliaskovic's user avatar
37 votes
3 answers
3k views

If $(A-B)^2=AB$, prove that $\det(AB-BA)=0$.

Let $A,B\in M_{n}(\mathbb{Q})$. If $(A-B)^2=AB$, prove that $\det(AB-BA)=0$. I considered the function $f:\mathbb{Q}\rightarrow \mathbb{Q}$, $f(x)=\det(A^2+B^2-BA-xAB)$ and I obtained that: $$f(0)=\...
ztefelina's user avatar
  • 2,086
37 votes
1 answer
1k views

Can some proof that $\det(A) \ne 0$ be checked faster than matrix multiplication?

We can compute a determinant of an $n \times n$ matrix in $O(n^3)$ operations in several ways, for example by LU decomposition. It's also known (see, e.g., Wikipedia) that if we can multiply two $n \...
Misha Lavrov's user avatar
36 votes
4 answers
7k views

Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

Math people: In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times n}...
Stefan Smith's user avatar
  • 8,202
36 votes
2 answers
777 views

$3 \times 3$ matrix with determinant a large power of $2$

Here's a little curiosity I found. The following $3 \times 3$ matrix has entries that are distinct primes $< 100$ and its determinant is $2^{19}$. $$ \pmatrix{71 & 31 & 97\cr 61 & 67 &...
Robert Israel's user avatar
36 votes
1 answer
1k views

Prove this determinant identity combinatorially

This is for those of you who understand the Lindstrom-Gessel-Viennot lemma. I am looking for a proof of the following identity using paths and such: Let $A$ be an $n\times n$ matrix, and for $i,j\in\{...
wircho's user avatar
  • 1,431
35 votes
4 answers
20k views

Determinant of the Kronecker Product of Two Matrices

I'd like to know how can be shown that $\det(A \otimes B) = \det(A)^m \det(B)^n$ when $A$ and $B$ are square matrices of size $n$ and $m$ respectively and $\otimes$ represents the Kronecker product of ...
Jonas Gomes's user avatar
  • 3,099
34 votes
3 answers
3k views

Prove that $\det( M) \in \mathbb{Z}$

Let $\Gamma$ be a finite multiplicative group of matrices with complex entries. Let $M$ denote the sum of the matrices in $\Gamma$. Prove that det $M$ is an integer. Hint: square $M$ and use the ...
user avatar
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
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
34 votes
6 answers
2k views

Proof that $\text{det}(AB) = \text{det}(A)\text{det}(B)$ without explicit expression for $\text{det}$

Overview I am seeking an approach to linear algebra along the lines of Down with the determinant! by Sheldon Axler. I am following his textbook Linear Algebra Done Right. In these references the ...
Jagerber48's user avatar
  • 1,441
34 votes
0 answers
1k views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $$V= \begin{bmatrix} ...
trion's user avatar
  • 353
33 votes
7 answers
31k views

Intuition behind a matrix being invertible iff its determinant is non-zero

Question I have been wondering about this question since I was in school. How can one number tell so much about the whole matrix being invertible or not? I know the proof of this statement now. But I ...
Vishal Gupta's user avatar
  • 6,986
32 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
  • 5,515

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