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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0
votes
6answers
125 views

Is it mathematically correct to replace 1 with I?

For example, $3=3I$ $=3\begin{bmatrix} 1 & 0 \\ 0 &1 \\ \end{bmatrix}$ $=\begin{bmatrix} 3 & 0 \\ 0 &3 \\ \end{bmatrix}$ ...
8
votes
4answers
230 views
+50

Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$

Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that: $$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$ under the following ...
4
votes
1answer
87 views

Determine the eigenvalue of a real matrix

I think to this question for two days : Let $A$ be a $3\times3$ real matrix such that $\det(A) = 1$ and $A^{-1}= A^T$. Prove that one of the eigenvalues is equal to $1$. I used the fact that ...
33
votes
2answers
1k views

Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = \...
1
vote
0answers
11 views

A determinant associated to point sets in the plane

Consider $n$ distinct points in the plane $z_1, \ldots, z_n$. Form the matrix $D$ containing their squared distances as entries: $$ D_{ij} \ = \ |z_i - z_j|^2 \, . $$ Obviously, this matrix is ...
0
votes
2answers
450 views

Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
-3
votes
1answer
51 views

evaluation of determinant without expanding

If $\;\det \begin{pmatrix} a & x & x & x \\ x & b & x & x \\ x & x & c & x \\ x & x & x & d \end{pmatrix} =f(x)-xf’(x)$ where $f'(x)$ denotes ...
5
votes
1answer
12k views

Expressing the determinant of a sum of two matrices?

Can $$\det(A + B)$$ be expressed in terms of $$\det(A), \det(B), n$$ where $A,B$ are $n\times n$ matrices? # I made the edit to allow $n$ to be factored in.
0
votes
1answer
51 views

What is the determinant of exp(matrix)? [duplicate]

Given a square matrix $A$, form the Lie series of it, which is defined by: $$ e^A = I + A + \frac{1}{2} A^2 + \frac{1}{3!} A^3 + \cdots + \frac{1}{n!} A^n = \sum_{k=0}^\infty \frac{1}{k!} A^k $$ Is ...
4
votes
0answers
60 views

How many subsets of $n$ linearly independent binary strings of length $n$?

Let's consider binary words of length $n$ with elements {-1,1}. There are $2^n$ binary words of length $n$. Now let's consider a subset of $n$ such binary words. All possible subsets are $\binom{2^n}{...
2
votes
0answers
38 views

What is the meaning of cofactor expansion?

I understand how to perform a cofactor expansion in finding the determinant. Can you explain what this method is really capturing or what thinking leads us to use this method?
2
votes
2answers
3k views

How many leading principal minors are there for a $4 \times 4$ matrix?

How many leading principal minors are there for a $4 \times 4$ matrix? Please explain in detail. I know for a $3 \times 3$ matrix.
3
votes
2answers
905 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
12 views

Showing expression for $ [ \mathbf{a,b,c}] [ \mathbf{u,v,w}] $ where $ [\mathbf{x,y,z}] $ is the triple scalar product of vectors in $\mathbb{R}^3 $

Any hints/solutions to how I can show $$ [ \mathbf{a,b,c}] [ \mathbf{u,v,w}] = \begin{vmatrix} \mathbf{a.u} & \mathbf{a.v} & \mathbf{a.w} \\ \mathbf{b.u} & \mathbf{b.v} &...
0
votes
0answers
4 views

Determinant of complete signed graph [closed]

Complete signed graph is graph where each node is connected to all other nodes via positive or negative edges. Thus adjacency matrix $A$ of a complete signed graph $G$ is $$A_{ij}=\begin{cases} 1 \ \ ...
2
votes
0answers
64 views

Is there a concept of “Cross determinant”?

Suppose $A = \begin{bmatrix}a & b \\ c & d \\\end{bmatrix}$. The determinant of $A$ is $$\det A = ad - bc.$$ Suppose $B = \begin{bmatrix}e & f \\ g & h \\\end{bmatrix}$. Now one could ...
2
votes
1answer
185 views

is this matrix invertible

Is the following matrix invertible? $\left[ \begin{matrix} \sum\limits_{x=1}^{n}{1} & \sum\limits_{x=1}^{n}{x} & \sum\limits_{x=1}^{n}{{{x}^{2}}} & \cdots & \sum\limits_{x=1}^{n}{{...
1
vote
0answers
24 views

Can $\sum_{|I|=k}\det(G_I)^2$ be represented by the eigenvalues of $G$?

In the question, $G\in R^{n\times n}$ is the symmetric positive semi-definite (SPSD) matrix, $\det(\cdot)$ is the determinant of the matrix, $G_I$ is a principal submatrix of the SPSD matrix $G$, and $...
1
vote
1answer
142 views

Calculating determinant of matrix $n\times n$ [closed]

Given $$M := \mbox{diag} (1, 2, \dots, n) - n \, I_n + n \,1_n 1_n^T$$compute the determinant of $M$.
0
votes
1answer
56 views

Given integers $m,n$, find integers $a,b,c$ such that $a^3+b^3+c^3-3abc=m n$

With a³+b³+c³-3abc=m (m-random integer) And a³+b³+c³-3abc=n(n-another integer) How to find a³+b³+c³-3abc=mn(m and n are Co prime) I came across this in an online math contest. Hint was given as ...
-1
votes
2answers
52 views

Determinant of a $3 \times 3$ matrix [closed]

I have a matrix of order $3 \times 3$. When I take its determinant it give 1700 from all the rows and columns except row 2. I don`t know whats going on $$\begin{bmatrix}1&20&0\\0&0&10\...
0
votes
1answer
22 views

Find conditions on $x$ and $y$ which guarantee that one can locally solve the following for $u(x, y)$ and $v(x, y)$

My understanding of this question is that I need to show that the following equations can be solved where $u$ and $v$ can be written as a function of $x$ and $y$. $xu^2+yv^2=9$ $xv^2-yu^2=7$ I ...
1
vote
0answers
14 views

Sum of multiplication of non intersecting elements in a matrix

Is there any formula for sum of multiplication of non-intersecting elements in a matrix? For example for a $3\times 3$ matrix \begin{align*} \begin{bmatrix} a_{11} &a_{12} &a_{13}\\ a_{21} &...
1
vote
3answers
70 views

Prove by induction that $\det(A^T) = \det (A)$ [closed]

If $A$ is an $n\times n$ matrix then $\det(A^T) = det(A) $. Prove by induction that the matrix obtained by deleting the $i^{\rm th}$ row and $j^{\rm th}$ column of $A^T$ is the transpose of the ...
0
votes
1answer
30 views

Determinant of complex matrix from real and complex parts

Is is possible to find the absolute value of the determinant of a complex matrix M, given two real matrices A & B of the same dimension of M; where for N = 2, M would be: $\begin{matrix} a_{00}...
11
votes
7answers
3k views

If A is a 2x2 matrix, what is det(4A) in terms of det(A)

If $A$ is a $2\times2$ matrix, what is $\det(4A)$ in terms of $\det(A)$? This seems trivial, but I'm not sure exactly what they are asking. I'm guessing I have some matrix $A = \begin{bmatrix}a&...
1
vote
3answers
36 views

Using row operations to compute the following 3x3 determinant

Use row operations to compute the following determinant $\begin{bmatrix}3&3&-3\\3&4&-4\\2&-3&-5\end{bmatrix}$ I know how to easily compute the determinant using $i - j + k$ ...
1
vote
3answers
67 views

Computing the $4 \times 4$ determinant of a matrix

Compute the determinant of $$\begin{bmatrix}1&-2&5&2\\0&0&3&0\\2&-4&-3&5\\2&0&3&5\end{bmatrix}$$ by first expanding along the first row (at ...
2
votes
0answers
69 views

Determinant – graphical derivation of formula?

I know how to compute determinants and I'm familiar with the geometrical meaning of determinant as the scaling factor of a unit (point/square/cube/hypercube)'s area/volume by applying a linear ...
1
vote
1answer
27 views

Is there a simple way of calculating determiant of “reverse arrowhead” matrix?

I have some problems with finding determinant of the following matix. It seems easy but I keep getting something wrong. I have tried simple Laplace expansion on the first row, but I have a feeling I ...
4
votes
2answers
60 views

Prove an equality of complex matrixes

If $A \in M_2(\mathbb{C})$ a matrix so that $$\det\left(A^2 + A + I_2\right)=\det\left(A^2 - A + I_2\right)=3 \tag1$$ then $$A^2\left(A^2 + I_2\right)=2I_2. \tag2$$ I tried to use Cayley-Hamilton ...
1
vote
0answers
25 views

Proof Determinant of Block Matrix does not depend of a variable

I have the following matrix (called BRM): $ BRM = \begin{bmatrix} -A & A & \mathbb{0}_{3\times3} & B & \mathbb{0}_{3\times1} & \mathbb{0}_{3\times1} \\ -C &...
12
votes
2answers
3k views

Coefficients of characteristic polynomial of a matrix

For a given matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed by the columns and rows with indices from $J$. If the characteristic polynomial of $A$ is $x^n+...
-2
votes
0answers
46 views

Given det(A) = 2. Find the Determinant of this Matrix

I've run into a roadblock that my textbook doesn't seem to be able to help me with. I am not understanding how to solve these type of questions. I am assuming to receive the answer given, you do some ...
9
votes
2answers
257 views

Determinant of a $2\times 2$ 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 $A$ ...
1
vote
0answers
35 views

Prove that the determinant of a triangular matrix only has one non zero permutation term

Using permutations explain how for a triangular matrix only one term can be non zero. Please do not include any proofs using the cofactor method. Edit (OP's attempt as written in the comment ...
2
votes
4answers
44 views

Find the determinant using colum or row operations

I find problem in simplification. When I tried to simplify I ended up doing the regular process of finding the determinant value. The matrix is $\begin{pmatrix} 1 & 1 & 1 \\ a & b & c \...
2
votes
1answer
42 views

To distinguish among the various subsets of $M_n(\Bbb R)$

I am having problem in doing a certain type of problems relating to matrices: To distinguish among the various subsets of $M_n(\Bbb R)$ such as symmetric, diagonal, diagonalizable, upper triangular, ...
9
votes
1answer
618 views

Idiotic determinant mistake?

I need to calculate $$\begin{vmatrix} \lambda & -1 & 0 & 0\\ -1 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & -1 \\ 0 & 0 & -1 & \lambda \end{vmatrix}.$$ For ...
1
vote
2answers
58 views

Finding the determinant of the 5x5 matrix but can't put it in lower triangular form

How to find the determinant of this 5x5 matrix? I can't put it in Lower or Upper Triangular form so I'm confused. I dont really know how to use laplace expansion $\begin{bmatrix}3&0&0&3&...
7
votes
3answers
155 views

How much can we tell about $\det(X)$ if we know $\det(I + X)$?

What can we tell about $\det(X)$ if we know $\det(I + X)$? Will it give some kind of bound for $\det(X)$? In general, if we know the determinant of matrix $A + X$, where $A$ is a constant matrix, how ...
1
vote
3answers
52 views

How to find the determinant of the matrix?

if $\det \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} = 1$ then $\det \begin{bmatrix}a - 6g&7b - 42h&c - 6i\\d&7e&f\\g&7h&i\end{bmatrix} = ?$ ...
-1
votes
1answer
31 views

Calculating the eigenvalues [closed]

I'm trying to understand the dynamics of the eigenvectors and the eigenvalues. My question is about formula for finding the eigenvalues. At 4:15(the athor starts the calculating at 1:30) of the given ...
8
votes
4answers
120 views

Matrix equation $A^2+A=I$ when $\det(A) = 1$

I have to solve the following problem: find the matrix $A \in M_{n \times n}(\mathbb{R})$ such that: $$A^2+A=I$$ and $\det(A)=1$. How many of these matrices can be found when $n$ is given? Thanks in ...
1
vote
1answer
32 views

Using determinants to find a recursive sequence

I am trying to compute a three diagonal determinant in order to find the recursive relation. Let $\Delta_{n}$=$\begin{vmatrix} 11 & 3 & 0 & 0 & \dots & 0\\ 13 & 11 & 3 &...
2
votes
1answer
60 views

How to find the determinant of a 5 x 5 matrix?

Finding a 3x3 matrix is easy, but how can I find the determinant of this 5x5 matrix?? I just need an example of the first couple steps to mimic $A =$ $\begin{bmatrix} 7&1&9&-4&3\\...
0
votes
1answer
54 views

Prove that determinant of a 2x2 symmetric positive definite matrix is positive by “completing the square” method.

From my understanding, determinant = product of Eigen values. Since it is a positive definite matrix, the eigen values are positive and hence, the determinant is ...
2
votes
2answers
84 views

$\det{\begin{bmatrix} A & B \\ C & D\end{bmatrix}}=(D-1)\det(A) +\det(A-BC) $? [duplicate]

I found this identity in Wikipedia, When $D$ is a $1 \times 1$ matrix, $B$ is a column vector, and $C$ is a row vector, then $$\det{\begin{bmatrix} A & B \\ C & D\end{bmatrix}}=(D-1)...
5
votes
1answer
58 views

Prove a matrix expression leads to an invertible matrix?

I want to prove matrix $C$ is invertible: $$C=I-A^TB(B^TB)^{-1}B^TA(A^TA)^{-1},$$ where $I$ is an identity matrix of appropriate dimensions, and $(A^TA)^{-1}$ and $(B^TB)^{-1}$ imply both $A$ and $B$ ...
1
vote
1answer
58 views

How to show the matrix $\left( \binom{x-i}{j-1}\right)_{1\leq i,j\leq 2r+1}$ has determinant (-1)^r and it's inverse?

After playing around in mathematica, I found that the matrix $\left( \binom{x-i}{j-1}\right)_{1\leq i,j\leq 2r+1}$ has determinant $(-1)^r$ for the first few $r$'s. How can I prove this this, or at ...