# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $m \times m$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $i,j\in\{... • 1,431 35 votes 4 answers 20k views ### Determinant of the Kronecker Product of Two Matrices I'd like to know how can be shown that$\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$represents the Kronecker product of ... • 3,099 34 votes 3 answers 3k views ### Prove that$\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$and use the ... 34 votes 1 answer 121k views ### Using the Determinant to verify Linear Independence, Span and Basis Can the determinant (assuming it's non-zero) be used to determine that the vectors given are linearly independent, span the subspace and are a basis of that subspace? (In other words assuming I have a ... • 773 34 votes 1 answer 12k views ### Proof that determinant rank equals row/column rank Let$A$be a$m \times n$matrix with entries from some field$F$. Define the determinant rank of$A$to be the largest possible size of a nonzero minor, i.e. the size of the largest invertible square ... • 12k 34 votes 6 answers 2k views ### Proof that$\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 ...
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