# 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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### 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 ...
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### 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 $\operatorname{adj}(A)$?
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### 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 ...
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### 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, ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Log-Determinant Concavity Proof

Can you please help me understand how he gets the equation marked by red from the above one ?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 $\det\left(\operatorname{adj}(A)\right) = \det(A)^{n-1}$. ...
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### 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 ...
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### 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 ...
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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 $\mathbb{F}$, $$\det(\mathbf{A}... 63 votes 1 answer 126k views ### Effect of elementary row operations on determinant? 1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ... • 3,941 60 votes 3 answers 20k views ### What is the origin of the determinant in linear algebra? We often learn in a standard linear algebra course that a determinant is a number associated with a square matrix. We can define the determinant also by saying that it is the sum of all the possible ... • 1,093 53 votes 2 answers 7k views ### Development of the Idea of the Determinant While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better ... • 2,700 24 votes 1 answer 6k views ### Invertible matrices over a commutative ring and their determinants Why is it true that a matrix A \in \operatorname{Mat}_n(R), where R is a commutative ring, is invertible iff its determinant is invertible? Since \det(A)I_n = A\operatorname{adj}(A) = \... • 13.1k 19 votes 4 answers 14k views ### Why is it true that \mathrm{adj}(A)A = \det(A) \cdot I? This is a statement in linear algebra that I can't seem to understand the proof behind. For a square matrix A, why is:$$\mathrm{adj}(A)A = \det(A) \cdot I$$Any explanation would be greatly ... • 1,803 14 votes 7 answers 10k views ### \operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A How can I prove that \operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A, if A and B are any two n\times n-matrices. Here, \operatorname{adj} A means the adjugate of the ... • 149 14 votes 4 answers 3k views ### Determinant of a specific circulant matrix, A_n Let$$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]...
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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 ...