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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7
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191 views

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 ...
0
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1answer
15 views

Inverse of matrix sum

I found on the Wikipedia page "Determinant" the following property: For any invertible $m \times m$ matrix $X$, $\det(X + AB) = \det(X) \det(I_m + BX^{-1}A)$. Is this true? If so, how is this ...
2
votes
1answer
284 views

Prove that the determinant of $ A^{-1} = \frac{1}{det(A)} $- Linear Algebra

If I have a single matrix A that is non-singular, how can I prove the determinant of its inverse = $\frac{1}{\det(A)}$? Prove: $$ \det(\mathbf{A^{-1}}) = \frac{1}{\mathbf{\det(A)}} $$ I know that ...
4
votes
1answer
46 views

Uniqueness of determinant

In Artin Algebra 2nd edition page 22, the author proved the uniqueness of determinant by saying that any matrix $A$ can be written in reduced row-echelon form $A'$: $A'=E_1\cdots E_kA$ where $E_i$ are ...
0
votes
0answers
18 views

Eigenvalues of (restrictions of) the standard representation of $S_n$

Let the permutation group on $n$ elements $S_n$ act on a set $S$ of size $k < n$ via permutations. Fix some ordering on the elements of $S$ to make this sensible. Is there any way to understand ...
0
votes
1answer
21 views

Finding the real irrational root of a cubic polynomial?

I just wanted to check if anyone can see a simpler way to solve this. Because I am not looking forward to using the cubic formula to solve it! $$ det(\lambda-AI) = \left| \begin{array}{ccc} \lambda + ...
0
votes
0answers
11 views

Proof for Determinants using Laplace and induction.

Matrix $A = (a_{ij}) \in M (n x n, Field)$, Matrix $B = ((-1)^{i+j}a_{ij})$ I need to prove that det(A)=det(B). I thought induction might be one solution, but I don't know how to apply the Laplace ...
0
votes
0answers
22 views

Reference for the proof of interlacing of eigenvalues of submatrices

If one has a $n \times n$ Hermitian matrix $A$ and one removes $k$ of the rows and their corresponding columns then the eigenvalues of the remnant interlace the eigenvalues of the full matrix. Can ...
-1
votes
1answer
32 views

Prove the equality of two determinants. [on hold]

Matrix $A = (a_{ij}) \in M (n x n, Field)$, Matrix $B = ((-1)^{i+j}a_{ij})$ Proof that $det(A)=det(B)$? Thanks in advance.
-4
votes
0answers
22 views
0
votes
1answer
23 views

AB = Identity matrix; matrices; determinants; proof

Let $M(n\times n, \mathbb Z)$ be the set of all $n\times n$- matrices with integer coefficients, and a matrix $A \in M$. Proof, that: There is exactly one matrix $B \in M(n\times n, \mathbb Z)$ with ...
4
votes
3answers
70 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 ...
2
votes
1answer
28 views

How to prove this result using Permutations? [on hold]

Let A be the set of all $3*3$ skew symmetric matrices whose entries are either -1, 0 or 1. If there are exactly 3 zeroes, three 1's and three (-1)'s, then prove that only 8 such matrices can exist.
1
vote
1answer
35 views

Determinant of $\lambda I + A^TA$

What properties $\lambda I + A^TA$ have? I know that $A^T A$ is positive semi-definite, and symmetric. I want to show that the determinant of $\lambda I + A^TA$ decreases as $\lambda$ increases!
0
votes
2answers
27 views

Determinant by nullifying

I am supposed to calculate the value for the determinant of this matrix. I didn't know what to do, so I looked up for the sample solution, which I don't understand. $$\left|\begin{array}{ccc} 18 ...
1
vote
0answers
28 views

Determinant over $\mathbb{C}$ of an $\mathbb{H}$-linear mapping.

Let $V = \mathbb{C}^n$ and let let $u$ be a $\mathbb{C}$-linear endomorphism of $V$. Then $u$ can also be considered as an $\mathbb{R}$-linear mapping $u_{\mathbb{R}}$. It is well known that $$\det ...
2
votes
2answers
62 views

Block Matrix Determinant Proof

I am trying to solve the determinant of a Block matrix $$\begin{bmatrix}A-Ia&B\\B &A-Ib \end{bmatrix}$$ where a and b are integers and I is an identity matrix, A and B are square. ...
0
votes
0answers
20 views

How to prove that a matrix with specific property is invertible?

If we have a square matrix $$ M = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & ...
2
votes
0answers
22 views

Log concavity/convexity of a determinant

I was wondering if anyone would be able to help me determine whether the following quantity is log concave or not with respect to $\alpha$? $$\left[\det(\textbf Y^\top \textbf P \textbf G \textbf ...
2
votes
1answer
51 views

Find the value of the Determinant

If $a^2+b^2+c^2+ab+bc+ca \le 0\quad \forall a, b, c\in\mathbb{R}$, then find the value of the determinant $$ \begin{vmatrix} (a+b+2)^2 & a^2+b^2 & 1 \\ 1 & (b+c+2)^2 ...
1
vote
2answers
29 views

How many solutions exist for a matrix equation $A^2=I$?

Let $A$ be a square matrix of order three or two, and $I$ be a unit matrix. How many solutions are possible for the equation $$A^2=I$$? In case the solutions are infinite, or very large, how do I ...
1
vote
1answer
41 views

Determinant and matrix power

I was wondering if there is a relation between the determinant of a matrix and the determinant of its powers. I mean I am looking for something like $$ \det (A^k) = f(\det(A), k). $$ A few check I ...
0
votes
0answers
18 views

Transformation into a field with the result being a multiple of the determinant

Let $K$ be a field, $n \in N$ and d: $M_{n,n}(K) \to K $ an homogeneous and skew invariant transformation where $M_{n,n}(K)$ are the matrices over the field. Show that there's a $d$ with $d = c * ...
0
votes
1answer
24 views

Given $\det(A)$ and $\det(B)$, is my calculation of $\det(-2B^T B A)$ correct?

Suppose $A$ and $B$ are $3 \times 3$ matrices with $\det(A) = -2$ and $\det(B) = -1$. What is the determinant of $C = -2 B^T B A$? I know that $$\det(A^T) = \det(A) \qquad \det(AB) = \det(A) ...
0
votes
1answer
69 views

Determinant of complex bordered matrix

Let $A$ be an $n\times n$ invertible matrix. Let $a$ be a complex number, let $\alpha$ be a row $n$-tuple of complex numbers and let $\beta$ be a column $n$-tuple of complex numbers. Show that ...
1
vote
2answers
131 views

Trying to find $\det (B)$

Let $A=\left(\begin{matrix}1&1&-1\\-1&1&1\\1&-1&1\end{matrix} \right)$,and ${A}^{T}B{\left( \cfrac{1}{2}{A}^{T}\right)}^{T}-8{A}^{-1}B=I$, How to compute $\left|B \right|$?。
1
vote
1answer
30 views

Determinant of 3 points.

I have $P=(p_1,p_2)$ and $Q=(q_1,q_2$) two points in $\mathbb R^2$, $P\ne Q$, and $R=(r_1,r_2)$ another point. What means the following determinant? $$\Delta (P, Q, R)= \begin{vmatrix} ...
1
vote
1answer
30 views

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = ...
5
votes
3answers
100 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\\ ...& ... & ... &... \\ ...
1
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1answer
35 views

Determinant of Hermitian Tridiagonal Matrix with Constant Upper and Lower Diagonals

I got this equation where the a terms are known but I want to determine a relationship between the b terms (so, no numerical analysis please). I know that the bi terms are real and the a terms are ...
0
votes
0answers
28 views

Determinant of matrix n x n [duplicate]

How to calculate $det\begin{bmatrix}1 & x_1 & x_1^2 \dots x_1^{n-1} \\ 1 & x_2 & x_2^2 \dots x_2^{n-1} \\ \\ 1 & x_n & x_n^2 \dots x_n^{n-1}\end{bmatrix}$?
1
vote
1answer
22 views

Determinant of 2 transpose matrix A and B.

Can you show me why $\det(A^T B^T) = \det(A)\det(B^T) = \det(A^T)\det(B)$ ? im really having a hard time finding its properties. i dont know what to search. please help.
0
votes
1answer
44 views

Correct proof of $\det(X+iY) \det(X-iY)>0$?

Can someone please look over my proof below as to why $\det(X+iY) \det(X-iY)>0$ for real matrices $X,Y$, such that $\det(X+iY)$, $ \det(X-iY)$ not both the zero, and tell me if it's correct ? My ...
2
votes
2answers
29 views

How to prove this result?

Let {$\Delta_1,\Delta_2,\Delta_3\cdots\cdots\cdots\cdots\Delta_n$} be the set of all determinants of order 3 that can be made with the distinct real numbers from set $S=\{1,2,3,4,5,6,7,8,9\}$. Then ...
1
vote
3answers
54 views

Determinant as a number that tells if a system has solution or not

There are many ways to define and interpret determinants. The one I'm more interested right now is the one that better describes its name: a number that can determinate if a system of linear equations ...
3
votes
1answer
66 views

Determining $\det(\mathbf{A})$ using the characteristic polynomial

Let the 3x3 matrix be $ \mathbf{A} = \begin {bmatrix} 3&1&0\\1&3&0\\0&0&1 \end {bmatrix}$. a) Determine its eigenvalues and eigenvectors. b) Do the eigenvectors ...
0
votes
2answers
114 views

Proving that $\det (A^2 - I) < 0 \Rightarrow \lambda \in (-1,1)$

Let $A$ be real square matrix. If $\det (A^2 - I) < 0$, then $A$ has an eigenvalue $\lambda \in (-1,1)$. How to prove this?
4
votes
1answer
31 views

linearly independent and determinant

This question says a matrix $\begin{bmatrix}a & b\\c & d\end{bmatrix}$ where $a_{ij}$ are real numbers. I need to prove that $\det|A|=ad-bc\neq0 \iff $the columns are linearly independent. ...
7
votes
1answer
84 views

Is there a deeper meaning behind the “determinant” formula for the cross product?

We all know that for all vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R^3}$, if $(a_x,a_y,a_z)^\top$ is the component form of $\mathbf{a}$ and similarly $(b_x, b_y, b_z)^\top$ is the component form of ...
0
votes
1answer
12 views

Express the vector $b=2i-3j+5j$ in terms of these set of three vectors

The three vectors are: $$a_1=i+j+k$$ $$a_2=i-j$$ $$a_3=i+j-2k$$ I have been asked to express the vector $b=2i-3j+5j$ in terms of the three vectors above like: $b= \alpha a_1+\beta a_2+\gamma a_3$. ...
0
votes
1answer
27 views

What method is used to find the determinant of this $4 \times 4$ matrix?

This is a pre-solved example in my book, I don't understand how they solved it. What method is used? Find the determinant of $A = \begin{bmatrix} 0 & 1 & 0 & 2\\[0.3em] -1 ...
2
votes
0answers
44 views

Determinant of a sum

We have that: $\textbf Y \in \mathbb{R}^{n \times q}, \textbf G \in \mathbb{R}^{n \times n}, \textbf P \in \mathbb{R}^{n \times n}, \textbf Q \in \mathbb{R}^{q \times q}$. Furthermore, $\textbf G$ is ...
3
votes
1answer
30 views

Find the value of the expression-

Consider a matrix $A=\begin{bmatrix}3 & 1\\-6 & -2\end{bmatrix}$, then $(I+A)^{99}$ equals ? So how can I expand this ? The solution paper gives the answer as $I+(2^{99}-1)A$
2
votes
1answer
45 views

How to prove these two statements?

Let A,B,C,D be real matrices (not necessarily square) such that $$A^T=BCD$$$$B^T=CDA$$$$C^T=DAB$$$$D^T=ABC$$ For the matrix S=ABCD, prove that $$S^3=S$$ and $$S^2=S^4$$ My little brother got this in ...
1
vote
1answer
48 views

Which of the following cannot be the value of $g(x)$

Let A = $\begin{bmatrix}1 & \tan x\\-\tan x & 1\end{bmatrix}$ then let us define a function $f(x)=\begin{vmatrix}A^{T}A^{-1}\end{vmatrix}$ then which of the following cannot be the value of ...
2
votes
1answer
37 views

Maximum value of $f(x) = \log_{(\tan x + \cot x)}(\det A)$ for a diagonal matrix $A$

If $$A =\begin{pmatrix} d_1 & 0 & 0 & 0 \\ 0 & d_2 & 0 & 0\\ 0 & 0 & d_3 & 0\\ 0 & 0 & 0 & d_4\\ \end{pmatrix}$$ ...
1
vote
1answer
29 views

$a_{ij}=i $ if $i+j=n+1$ and $0$ otherwise; compute det $A$

The entries of the matrix is specified by this rule, $A=(a_{ij})\in M_n(\mathbb R)$, $a_{ij}=i$ if $i+j=n+1$ and $0$ otherwise. Compute det $A$ > I have seen ...
2
votes
2answers
40 views

Easiest way to calculate determinant 5x5 witx x

I would like to calculate this determinant: ...
1
vote
2answers
457 views

The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
1
vote
1answer
37 views

The value of $\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}$ for given determinant is

If $a\neq p$, $b\neq q$, $c\neq r$ and $\left|\begin{array}{cc}p&b&c\\a&q&c\\a&b&r \end{array}\right|= 0$ then the value of $\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}$ is (a) ...