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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6
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1answer
223 views

A particular (functional) determinant calculation

One wants to calculate the quantity, $\det'(\frac{\partial}{\partial t} - i [\alpha, ])$ where the prime on the "det" means that one wants to do a product over only non-zero eigenvalues of the ...
6
votes
3answers
270 views

Which is easier to work out: determinant or inverse?

Suppose $A\in M_n(R)$ be a $n\times n$ matrix over some ring $R$. Which of the following two tasks is easier? to work out $\det(A)$; to work out $A^{-1}$. More specifically, I want to know the ...
6
votes
1answer
122 views

Can we determine the determinant?

Could someone prove that this determinant is not zero? $$\left| \begin{array}{cccc} 1^n & 2^n & \cdots & (n+1)^n \\ 2^n & 3^n & \cdots & (n+2)^n \\ ...
6
votes
0answers
185 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha ...
6
votes
1answer
183 views

What does abstract algebra have to say about the determinant?

The determinant is a homomorphism from the multiplicative monoid of matrices to the multiplicative monoid of a field (right?). I find this to be the most intuitive way to interpret some of the ...
6
votes
1answer
150 views

Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
6
votes
0answers
402 views

Determinant of Transpose of Linear Map

I'm trying to find a way to prove that the determinant of the transpose of an endomorphism is the determinant of the original linear map (i.e. det(A) = det(Aᵀ) in matrix language) using Dieudonne's ...
6
votes
1answer
104 views

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 $A$. ...
5
votes
5answers
695 views

Is a square matrix whose diagonal and antidiagonal elements are all zero always singular?

Consider an $n\times n$ matrix whose primary and secondary diagonal elements are all zero. Does it necessarily follow that the determinant vanishes for these matrices? When $n=1,2,3,4$, the matrix is ...
5
votes
3answers
900 views

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
5
votes
8answers
1k views

How to calculate the following determinants (all ones, minus $I$)

How do I calculate the determinant of the following $n\times n$ matrices $ \left[ \begin {matrix} 0 & 1 & \ldots & 1 \\ 1 & 0 & \ldots & 1 \\ \vdots & \vdots & ...
5
votes
2answers
909 views

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 ...
5
votes
4answers
280 views

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 ...
5
votes
4answers
210 views

Linear Algebra: different determinant answers

I'm having a problem verifying my answer to this question: Solve for x: $$\left| \begin{array}{cc} x+3 & 2 \\ 1 & x+2 \end{array} \right| = 0$$ I get: $(x+3)(x+2)-2=0$ $(x+3)(x+2)=2$ ...
5
votes
4answers
7k views

Determinant of a block matrix [duplicate]

How prove this equality for a block matrix: $$\det\left[\begin{array}[cc]\\A&C\\ 0&B\end{array}\right]=\det(A)\det(B)$$ I tried to use a proof by induction but I'm stuck on it. Is there a ...
5
votes
4answers
473 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 ...
5
votes
3answers
884 views

Determinant of a linear transformation defining matrix transpose

So if I define a linear transformation $ T: M_{n\times n}(R) \rightarrow M_{n\times n}(R) $ and $ T(A)=A^t $ what would be its determinant?
5
votes
4answers
323 views

Decompose this matrix as a sum of unit and nilpotent matrix.

Show that the matrix $A=\begin{bmatrix} 1 & 0 \\ 2 & 1 \\ \end{bmatrix}$ can be decomposed as a sum of a unit and nilpotent matrix. Hence evaluate the matrix ...
5
votes
4answers
273 views

Question about determinants

I am working on some practice problems and I'm unsure where to begin this problem. It starts off by giving $\det(X)= 1$ for the following matrix $X$:$$ \begin{matrix} a & 1 & d \\ b & 1 ...
5
votes
4answers
592 views

Vector space of polynomials over $\mathbb{R}$ with degree $\leqslant n-1$

Let $P \in \mathbb{R}_{n-1}[X]$ be a polynomial of degree $n-1 \geqslant 0$. Let $\mathbb{R}_{n-1}[X]$ be the vector space of polynomials with degree $\leqslant n-1$ over $\mathbb{R}$. Show ...
5
votes
3answers
445 views

Special orthogonal matrices have orthogonal square roots

Let $A$ be an orthogonal matrix with $\det (A)=1$. Show that there exists an orthogonal matrix $B$ such that $B^2=A$. Thank you very much.
5
votes
1answer
866 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 $\det(A)I_n = A\operatorname{adj}(A)$ then I ...
5
votes
3answers
88 views

Determinant of the inverse matrix [duplicate]

I'm seeking for a proof of the following: Let $A$ be an invertible matrix. Then the determinant of $A^{-1}$ equals: $$\left|A^{-1}\right|=|A|^{-1} $$ I don't know where to begin the proof. Any ...
5
votes
7answers
1k views

Proving the relation $\det(I + xy^T ) = 1 + x^Ty$

Let $x$ and $y$ denote two $n$ - length column vectors. Prove that $$\det(I + xy^T ) = 1 + x^Ty$$ Is Sylvester's determinant theorem an extension of the problem? Is the approach same?
5
votes
3answers
106 views

How to show that $\det(A+I)\ne 0$

How to show that for any skew symmetric real matrix $A$, we have $\det(A+I)\ne 0?$ Where to begin? I'm looking for some clue only.
5
votes
4answers
720 views

Linear Algebra - four “true or false” questions about matrices and linear systems

I'm reviewing for my linear algebra course, and have four "true or false" questions that I'm struggling to prove. I've included my approach to the solutions in brackets below them: 1) If $A^2 = B^2$, ...
5
votes
4answers
2k views

Intuition behind Matrix being invertible iff determinant is non-zero

I have been wondering about this question since I was in school. How can one number tell so much about the whole matrix being invertible or not? I know the proof of this statement now. But I would ...
5
votes
5answers
465 views

How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$

Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix $$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$ How to prove that ...
5
votes
3answers
300 views

computing determinant of a matrix

let $A$ be an $n\times n$ matrix with entries $a_{ij}$ such that $a_{ij}=2$ if $i=j$. $a_{ij}=1$ if $|i-j|=2$ and $a_{ij}=0$ otherwise. compute the determinant of $A$. using the famous formula ...
5
votes
3answers
84 views

Prove that $\det(AA^T+I)\ge 1$

If $A$ is a matrix with real entries, prove that $$\det(AA^T+I)\ge 1.$$ I tried using the eigenvalues. One thing came into my mind: maybe $AA^T$ is positive definite (I don't know whether this is ...
5
votes
3answers
664 views

Determinants of block matrices

Let $A,B \in \mathbb{R}^{n,n}$. Now $C = \begin{pmatrix} A & iB \\ -iB & A \end{pmatrix}$ and $D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$. Show that $\det(C) \in \mathbb{R}$ ...
5
votes
2answers
260 views

Finding the determinant of a matrix

I am given $A = \begin{pmatrix} a & b\\ c & d \end{pmatrix} $ and B = $ \begin{pmatrix} e & f\\ g & h \end{pmatrix}$ whose elements are non-zero reals. If $BA = I$, where ...
5
votes
2answers
132 views

Determinant of $2\times 2$ Block Matrix

I would like to know the proof for: The determinant of the block matrix\begin{pmatrix} A & B\\ C& D\end{pmatrix} equals $(D-1) \det(A) + \det(A-BC) = (D+1) \det(A) - \det(A+BC),$ when $A$ is a ...
5
votes
2answers
713 views

Is the determinant of a zero divisor zero?

Suppose that $A$ is a zero divisor in the ring of $(n\times n)$-matrices over the ring $R$. Is $\det(A) =0$ if $R$ is a field? Is $\det(A) =0$ if $R$ is an integral domain? It's not necessarily ...
5
votes
1answer
154 views

Determinant evaluation for matrix with $-1, 2, -1$ below/on/above diagonal [duplicate]

What is the trick for evaluating the determinant of this matrix? $$\begin{bmatrix} 2 & -1 \\ -1 & 2 & -1 \\ & -1 & 2 & -1 \\ && -1 & 2 & -1 \\ &&& ...
5
votes
3answers
111 views

Find the determinant of the following matrix

Find the determinant of the following matrix: $$A = \begin{bmatrix} 1+x_1^2 &x_1x_2 & ... & x_1x_n \\ x_2x_1&1+x_2^2 &... & x_2x_n\\ ...& ... & ... &... \\ ...
5
votes
2answers
361 views

Determinant of rank-one perturbation of a diagonal matrix

Let $A$ be a rank-one perturbation of a diagonal matrix, i. e. $A = D + s^T s$, where $D = \DeclareMathOperator{diag}{diag} \diag\{\lambda_1,\ldots,\lambda_n\}$, $s = [s_1,\ldots,s_n] \neq 0$. Is ...
5
votes
1answer
117 views

Does the identity $\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2$ hold for $g \in U(n)$?

In a paper (corollary 1, p.14) the following identity is used: Let g be a unitary matrix. Then: $$\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2 \text{ for }g \in U(n)$$ Now my question is why this ...
5
votes
1answer
222 views

What is $\frac{\det(A+tI)}{\det(B+tI)}$ as $t\to0$?

If $A$ and $B$ are two real $2\times 2$ matrices with $\det A = 0 $ and $\det B = 0 $ and $\mathrm{tr}(B)$ is non zero. then what will be limit of $$\lim_{t\to0}\frac{\det(A+tI)}{\det(B+tI)}$$ I used ...
5
votes
1answer
67 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
5
votes
1answer
63 views

Determinant of a matrice $a_{ij}=e^{a_ib_j}$

1) Let $a_1<\dots<a_n$ real numbers and $\lambda_1,\dots,\lambda_n\in\mathbb{R}\backslash\{0\}$ Let $f(x)=\lambda_1e^{a_1x}+\dots+\lambda_ne^{a_nx}$ Show that $f$ has at most $n-1$ zeroes 2) ...
5
votes
1answer
64 views

Determinant of a $n\times n $ matrix

Let $n$ be a positive odd integer and let $A$ be a symmetric $n\times n$ matrix of integer entries such that $a_{ii}=0,i=1,2.....n$. Show that the determinant of $A$ is even. I tried using ...
5
votes
1answer
91 views

Properties of determinants.

Is this property of a determinant true? $$|A^3| = |A|^3.$$ I haven't studied about this but while working out on a sum, wondered if this could be true, I'll check out on other sums too if this ...
5
votes
1answer
162 views

Multiplication of determinants

Show that for any vectors $\bf{a}$,$\bf{b}$,$\bf{c}$,$\bf{u}$,$\bf{v}$,$\bf{w}$ in $\mathbb{R}^3$, ...
5
votes
1answer
178 views

Use row reduction to prove that $\det(\mathbf{A})=\det(\mathbf{A}^{T})$

I need to prove that the determinant of a matrix is equal to the determinant of its transpose. This fact is obviously easy to prove using the definition of the determinant, but the question stipulates ...
5
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1answer
95 views

Most elementary proof that a determinant is divisible by $m$

So a challenge problem states that you have an $n \times n$ matrix, where each entry is an integer between $0$ and $9$, and when each row is read as a base-10 number the number is divisible by a ...
5
votes
1answer
241 views

Why is the formula for the determinant as it is?

I know this sounds like such a vague question, but seeing as we all know the formula for the determinant of a general nxn matrix, I want to know exactly why we define it as such. I know that ...
5
votes
1answer
109 views

Upper Triangular Block Matrix Determinant by induction

We want to prove that: $$\det\begin{pmatrix}A & C \\ 0 & B\\ \end{pmatrix}= \det(A)\operatorname{det}(B),$$ where $A \in M_{m\times m}(R)$, $C \in M_{m\times n}(R)$,$B \in M_{n\times n}(R)$ ...
5
votes
2answers
380 views

Determinant (and invertibility) of generalized Vandermonde matrix

I have stumbled upon the following generalization of Vandermonde matrix when solving some problem in linear algebra related to Jordan normal form. Let us consider some number $\lambda$ and we assign ...
5
votes
1answer
88 views

Determinants and cofactors?

My professor gave us this definition for determinants for a $n \times n$ matrix $A$: $$\det(A) = a_{11}C_{11} + a_{12}C_{12} ... + a_{1n}C_{1n} $$ where $C_{1j}$ is the cofactor of $A$ on $a_{ij}$. ...