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.

The determinant is a value that can be computed from the entries of a square matrix. This value is different from $0$ if and only if the matrix has an inverse and the determinant of the identity matrix is equal to $1$. For instance, for a $2\times 2$ matrix whose entries of the first line are $a$ and $b$ and whose entries of the second line are $c$ and $d$, the determinant is $ad-bc$.

If $f\colon\mathbb{R}^n\longrightarrow\mathbb{R}^n$ is a linear map and if $b$ is a basis of $\mathbb{R}^n$, then the determinant of the matrix of $f$ with respect to $b$ does not depend upon the choice of $b$; this number is called the determinant of $f$. The linear map $f$ has an inverse if and only if its determinant is not $0$.

Determinants are useful in the analysis of systems of linear equations and in the study of endomorphisms of finite-dimensional vector spaces.

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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 have an inverse. I can find the determinant of a…
Jamie Banks
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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 $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n \det \left(A^2+B^2+C^2\right)$$ This problem is…
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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 polynomial of the matrix $\det(A-\lambda I)$.…
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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 to me in plain English.
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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?
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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 undermined the entire answer. However, it can be…
goblin GONE
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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: $$\left| \begin{matrix} 2 & 3&0&2&8 \\ 3 &…
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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 the introduction to Chapter 4: The determinant,…
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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 entire matrix is filled. At the end, the…
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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 following parallelograms the same? $(1)$ parallelogram with…
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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 main diagonal all equal -1 and entries on even lower…
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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?
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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 of $1$'s Eigenvalues of the rank one matrix…
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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 for any dimensions?
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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 $M$?
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