# 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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### Definition of second cofactor of a matrix

I am reading a paper and I came across with the term a second cofactor of matrix $M$ Let $C_{ab}$ be the operator cofactor, that is $C_{ab}$(M) is the cofactor o matrix $M$ . Is this the definition ...
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### $\frac{1}{24xyz} ( {A}_{1}- {A}_{2})$ is equal to

Suppose $xyz \ne 0$ and ${A}_{1} =$ \begin{vmatrix} (1+2x)^{2} & (1+2y)^{2} & (1+2z)^{2} \\ (1+3x)^{2} & (1+3y)^{2} & (1+3z)^{2} \\ (1+4x)^{2} & (1+4y)^{2} & (1+4z)^{2} \...
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### Linear independence of vectors and minor of matrices

Prove that if a minor of order $k$ is nonzero, then the corresponding columns of the matrix are linearly independent "The rank of a matrix is the maximal order of a nonzero minor of $A$" ...
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### How to prove this equation about calculation of matrix determinant?

How to prove the equation about the determinant of Matrix $M$, i.e., $|M|=\frac{(M \cdot a) \times (M \cdot b) \cdot (M \cdot c)}{a \times b \cdot c}$ where $a$, $b$ and $c$ are arbitrary vectors. ...
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### Equality of ideals generated by matrix minors

Let $A,B\in M_n(R)$ matrices over commutative ring. We say that $A\sim B \Leftrightarrow B=PAQ$ where $P,Q$ are invertible. Denote by $\Delta_k(A)$ the ideal in $R$ which is generated by all the ...
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### Principal Vector Exercise from Barrett O'Neill textbook

Prove that $\left|\begin{array}{ccc}{{v}_{2}}^{2}& -{v}_{1}{v}_{2}& {{v}_{1}}^{2}\\ E& F& G\\ L& M& N\end{array}\right|=0$ then $\overset{\to }{v}$ is principal vector where ...
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### Determinant of product of matrix and vectors

In determinants, we have the property that $det(AB)=det(A)det(B)$, I believe this can also be extended to the product of three matrices i.e. $det(ABC)=det(A)det(B)det(C)$. Given $X$ is a vector of ...
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### Determinant of family of Toeplitz matrices. Can we use recursion?

When investigating another question regarding matrix let us call it $M_{10}$ I found a peculiar pattern which I can't prove. We can define $M_n$ to be the $n\times n$ Toeplitz matrix where the ...
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### Determinant of a Toeplitz matrix

How can I calculate the determinant of the following Toeplitz matrix? \begin{bmatrix} 1&2&3&4&5&6&7&8&9&10\\ 2&1&2&3&4&5&6&7&8&9 ...
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### $\det(I+A)=1+\operatorname{Tr}(A)$ if $\operatorname{rank}(A)=1$.

Let $A$ be a complex matrix of rank $1$. Show that $$\det (I+A) = 1 + \operatorname{Tr}(A)$$ where $\det(X)$ denotes the determinant of $X$ and $\operatorname{Tr}(X)$ denotes the trace of $X$. Any ...
### space of diagonal matrices with entries $e^{e^{s_1}},e^{e^{s_2}},\cdot\cdot\cdot$ s.t. $\sum_{i \ge1} s_i=0.$
Consider a diagonal matrix $A$ with entries $e^{s_1},e^{s_2},\cdot\cdot\cdot$ on the diagonal s.t. $\sum_{i \ge1} s_i=0.$ From what I understand this is a subgroup of $SL_n(\Bbb R).$ This is because ...
### Calculating determinant of $A^n$ given the matrix $A$
Find if $det(A)=det(A^n)$ for $n>1$. How do I tackle questions like this, in general if the matrix $A$ is provided in the question? Should I work out with the basic definition of a determinant, ...