# Tagged Questions

Question about determinants, computation or 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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### $\det\left(I + A^TA^{-1}\right) = 2\left(1 + \operatorname{tr}\left(A^TA^{-1}\right)\right)$

Let $A$ be an invertible $3\times3$ matrix with complex values. Prove that: $$\det\left(I + A^TA^{-1}\right) = 2\left(1 + \operatorname{tr}\left(A^TA^{-1}\right)\right)$$ I've tried to solve this ...
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### Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$

Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues. If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be ...
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### Existence of an $n\times n$ real matrix $A$ such that $A^2=-I$.

Let $A$ be a $n\times n$ real matrix $A$ such that $A^2=-I$. Such an $A$ cannot be, Orthogonal. Invertible. Skew-symmetric. Symmetric. Diagonalizable. I tried to figure out the answer by looking ...
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### Historical meaning and usage of determinant

Can anyone please explain how, why, and where determinants were developed/formalized? What was their historical usage? Why were they initially formulated and what were they used for (and later ...
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### Determinant bundle of a tensor product

Let $X$ be a ringed space (for example, a scheme or a manifold). If $V$ is a locally free $\mathcal{O}_X$-module of rank $n$, then $\mathrm{det}(V) := \Lambda^n V$ is a locally free $\mathcal{O}_X$-...
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### Determinant of matrices along a line between two given matrices

The question, with no simplifications or motivation: Let $A$ and $B$ be square matrices of the same size (with real or complex coefficients). What is the most reasonable formula one can find for ...
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### If all entries of matrix $X$ are the same, then $\det (A+X)\det (A-X) \leq \det (A^2)$

I want to prove that $\det (A+X)\det (A-X) \leq \det (A^2)$ where $X$ is a matrix whose $n^2$ entries are all the same. I tried to write down the expressions involved but that didn't help me prove ...
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### Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
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### Directional derivative of the determinant

Please help me find the mistake in my derivation: Let $f:M_{n,n}(\mathbb{R}) \to \mathbb{R}$ be the determinant function, $f(A)=det(A)$. Let $p_A(x)$ denote the charecteristic polynomial of $A$. ...
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### How to prove $I + t X$ is invertiable for small enough $| t | ?$

Let $X \in \text{GL}_n(\mathbb{R})$ be an arbitrary real $n\times n$ matrix. How can we prove rigorously: $$\underset{b>0} {\exists} : \underset{|t|\le b} {\forall} : \det (I + t X) \neq 0$$ If ...
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### Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$

Let $\bf u$ be any column vector and $\bf v$ be any row vector, each with $n \geq 2$ arbitrary entries from a field. Then it is well known that ${\bf u} {\bf v}$ is an $n \times n$ matrix such ...
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### Find the determinant of $A + I$

Given a real valued matrix $A$ such that $A$ satisfies $AA^T = I$ and $\det(A)<0$, calculate $\det(A + I)$ My start : Since $A$ satisfies $AA^T = I$, $A$ is a unitary matrix. The determinant ...
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### Is $\det(AB) =\det(BA)$

I am having trouble proving if $$\det(AB) = \det(BA)$$ is right or wrong. $A,B$ are square matrices. Can you please point me to the right direction? Thank you
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### Proof If $AB-I$ Invertible then $BA-I$ invertible.

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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### Why must the determinant of a matrix with with integer entries be an integer?

Why must the determinant of a matrix with integer entries be an integer? Note: I know what a determinant of a matrix is, not sure how to explain this question. Is that because if the matrix is made ...
### If $A^T=-A$, then A is not invertible
Let $n \in \mathbb{N}$ be odd and $A \in$Mat$(n,\mathbb{R})$ with $A^T=-A$. Show that $A$ is not invertible. I have no idea how to start this...
### Determinant of a matrix with $t$ in all off-diagonal entries.
It seems from playing around with small values of $n$ that  \det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 &...