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
2answers
38 views

Find the eigenvalues of the following matrix

Consider $A =\left( \begin{array}{ccc} -1 & 2 & 2\\ 2 & 2 & -1\\ 2 & -1 & 2\\ \end{array} \right)$. Find the eigenvalues of $A$. So I know the characteristic polynomial is: ...
3
votes
5answers
283 views

Is the determinant of this matrix positive or negative?

$\left( \begin{array}{ccc} 1 & 1000 & 2 & 3 &4\\ 5 & 6 &7&1000 &8\\ 1000&9&8&7&6\\ 5 & 4&3&2&1000\\ 1&2&1000&3&4\\ ...
2
votes
1answer
29 views

Every skew-symmetric matrix has a non-negative determinant

I'm breaking this up into the even case and odd case (if $A$ is an $n\times n$ skew-symmetric matrix). So when $n$ is odd, we have: $\det(A)=\det(A^T)=\det(-A)=(-1)^n\det(A)\Rightarrow ...
1
vote
1answer
40 views

Can -3 and 2 be eigenvalues of the following matrix?

Can $-3$ and $2$ be eigenvalues of and nxn matrix B such that $A = B^{2}+B-6I$ and A's determinant is $0$? So this is what I concluded: At first glance, it can be seen that the matrix $A$ can be ...
1
vote
0answers
27 views

Does the determinant of a complex-valued matrix have a geometric interpretation?

The determinant of a real-valued matrix can be seen as the volume of the parallelotope with the column vectors as the sides. Is there an analogous interpretation for complex-valued matrix ...
0
votes
4answers
53 views

Prove determinant is zero

If $M = \begin{vmatrix} 1 & a & b+c \\ 1 & b & a+c \\ 1 & c & a+b \\ \end{vmatrix}$ Show that M = 0 WITHOUT expanding the determinant. I ...
2
votes
1answer
28 views

Write the determinant as a polynomial expression in the elementary symmetric polynomials

How to write $\det\begin{bmatrix}x_1&x_2&x_3&x_4\\x_2&x_3&x_4&x_1\\x_3&x_4&x_1&x_2\\x_4&x_1&x_2&x_3 \end{bmatrix}$in terms of elementary symmetric ...
-1
votes
0answers
33 views

The set of matrices with nonnegative determinant is not a subspace. [on hold]

Disprove using a counterexample: The set of all $3\times 3$ matrices with determinant $\ge 0$ is a subspace of $M_3(\Bbb C)$.
0
votes
2answers
64 views

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____?

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____? This is a fill-in-the-blank problem that I found in my paper, but I don't have this answer.
0
votes
0answers
34 views

Find Eigenvalues of Infinite Matrix

I have the matrix $M$ acting on $l^2(\mathbb{N;C})$ given by the components $$ M_{n,n'} = V_n\delta_{n,n'} + A\delta_{n,n'+1}+A^\ast\delta_{n,n'-1} $$ where $V_n$ is real and obeys a periodicity ...
0
votes
1answer
17 views
0
votes
3answers
43 views

Evaluating a determinant for eigenvalues

I need to evaluate $$\left| {\matrix{ {3 - \lambda } & 1 & 1 \cr 2 & {4 - \lambda } & 2 \cr 1 & 1 & {3 - \lambda } \cr } } \right|$$ A direct computation ...
0
votes
2answers
32 views

Use row operation to find the determinant?

Use row operations to find the determinant: Can someone give me a full answer please? Also can anyone tell me if the sign of the determinant matters ? Row operations : Det ( e(A) ) = ...
1
vote
1answer
18 views

Determinant of the Kronecker product involving the identity

Let $A$ be a square matrix and $I$ the $k \times k$ identity matrix. Then the identity $$ \det(A \otimes I) = \det(A)^k,$$ holds as can be seen from a general result on the determinant of block ...
-1
votes
1answer
10 views

The maximum value of r?

A point $A = (a,b)$ is defined such that it lies on the graph $y = x^2 +1$ A point $B = (c,d)$ is defined such that it lies WITHIN the area of $ (x+2)^2 + (y+2)^2 = r^2$ Let's define a matrix $M = ...
-1
votes
0answers
23 views

Matrix determinant, eigenvalues [closed]

"...has the determinant $x*y*(1-ab)$. Since $L<bK<b(aL)$, $1-ab<0$ and the fixed point is a saddle. So I know that for the fixed point to be a saddle, one eigenvalue must be positive and one ...
-1
votes
2answers
16 views

Finding determinant of following matrix

I need to find determinant of following matrix . I did it by simply doing $R_5$ - $R_1$ . and then evaluating the determinant .But its a lengthy process but answer came out.. But another thing i have ...
-1
votes
1answer
20 views

To find the determinant in this question

Given $A$ by $4×4$ non singular matrix and $B$ be matrix obtained from A by adding to its third row twice the first row .Then $det(2A^{-1}B)$ is $A:2$ $B:4$ $C:8$ $D:16$ I cannot think anything ...
2
votes
2answers
36 views

Need help with determinant question

Can pls someone help me to understand rom how they have gone from first row in top determinant to first row in second determinant
0
votes
3answers
29 views

A quick way to generate 3x3 matrices with determinant equal to 1?

Perhaps a formula involving the row number and column number of an element or just some parametric equations for each element. I know that I can just multiply two of these matrices together to get ...
0
votes
0answers
19 views

Find the values of x,y,z so that the 3 x 3 matrix is singular?

Find the values of x, y, z that the matrix is singular? With an explanation.
0
votes
1answer
32 views

How to prove distributive property of a determinant?

How to prove that $|A\cdot B| = |A|\cdot|B|$ where A and B are square matrices of the same size? P.S.: This proof is not mentioned in my textbook, nor was I able to find it on the web.
2
votes
0answers
20 views

What is the simplest way to solve determinant of a $n \times n$ matrix by upper and lower triangular matrices?

I know the basic rules to solve for the determinant of an $n \times n$ matrix using upper and lower triangular matrices, but what is the simplest way?
3
votes
2answers
44 views

Proof that the characteristic polynomial of a $2 \times 2$ matrix is $x^2 - \text{tr}(A) x + \det (A)$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $C_{A}(x) := \det(xI-A)$ be the characteristic polynomial of A. Show that ...
1
vote
2answers
74 views

Is there a general form for the determinant of this matrix?

This came up in trying to deal with small oscillations of an $N$-pendulum. I obviously want to calculate the characteristic polynomial in $\omega^2$ to see if I can deal with the equation even in ...
1
vote
0answers
15 views

Trace of the exterior powers of linear operators

Given linear operators $K_1,\ldots,K_m$ on a Hilbert space $\mathcal H$, what can we say about the trace of their exterior product $Tr \,(K_1\wedge \cdots \wedge K_m)$ ? More precisely: 1) If we ...
1
vote
0answers
18 views

divergence form of the determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
1
vote
2answers
39 views

Let $A$ be a $3×4$ matrix. Estimate $\det(A'A)$ and $\det(AA')$

Let $A$ be a $3×4$ matrix. Estimate $\det(A'A)$ and $\det(AA')$. I would first assume that $A$ has rank $3$. Then $A'A$ would be a $4\times 4$ matrix with rank $3$ and therefore it would have ...
0
votes
1answer
34 views

A combinatorial coefficient linked to exterior product

I am looking at the following sum $$ \sum c_1\wedge \cdots\wedge c_n $$ where the summation ranges over $c_1,\ldots,c_n$ such that each $c_i\in\{a,b\}$ and $a$ appears exactly $j$ times. Thus, using ...
1
vote
1answer
25 views

Functions of several variables and $Df$

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth function and let $g:\mathbb{R}^n \rightarrow \mathbb{R}$ be defined by $g(x_1,...,x_n)=x_1^5+...+x_n^5$. Suppose $g\circ f\equiv 0$. Show that ...
2
votes
1answer
40 views

Invertibility of block matrices, with the property of being symmetric, positive definite, and of full rank:

If A and B are real matrices, with A being symmetric, B having at least as many columns as rows, and the matrix C defined as: $$ \begin{bmatrix} A & B^T \\ B & 0 \\ ...
0
votes
3answers
66 views

How do I find the determinant of a 4x4 matrix when the diagonal is made up of variables? [closed]

Evaluate: $\det(A)$, where $A= \begin{bmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1 \\ 1 & 1 & a & 1 \\ 1 & 1 & 1 & a\end{bmatrix}$
0
votes
2answers
43 views

Determinants of $3\times3$ matrices with full rank

I have two $3\times3$ matrices $A$ and $B$ where $$A = [c_1 : c_2 : c_3]$$ $$B = [c_1 : c_1 + c_2 : c_1+c_2+c_3]$$ where $c_i$ is the $i^{th}$ column of $A$. Given that $|A| = 1$, I am to find the ...
0
votes
1answer
41 views

$2X2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors

Give an example of $2X2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors. I would like to know a systematic answer of how to get this. My guess ...
3
votes
3answers
254 views

For which $x$ is the determinant vanishes?

For which values of $x \in \mathbb{R}$ does the determinant of the matrix $$ M = \begin{pmatrix} x & 0 & 1 & 2 \\ 2 & x & 0 & 1 \\ 1 & 2 & x & 0 \\ 0 & 1 ...
0
votes
0answers
41 views

matrix determinant changes when doing row operation, so weird O_o

To calculate the determinant of a matrix, you can subtract a row by another, and the determinant will not change. However, in the following matrix, the determinant is -2. \begin{bmatrix} 1 ...
0
votes
0answers
10 views

Straight Line equation from determinant of a matrix

Question: det(matrix{{2, r, y}, {n, 1, 1}, {2, 1, 3}}) = 0 if the gradient or m is 4, what is the straight line equation? Steps using diagonals method: ...
2
votes
0answers
47 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
1
vote
1answer
54 views

Cramers Rule. The why and how.

Can someone explain how Cramer's rule works. I understand the mechanics of it, and it's fairly straightforward to show algebraically that it's equivalent to GJ and substitution, but what's happening ...
0
votes
2answers
60 views

Fields over which a matrix is not invertible

I am trying to find the fields over which the matrix: $\left(\begin{matrix} 1 & 2 & 3 \\ 0 & -1 & 2 \\ 1 & 0 & -2 \end{matrix}\right) $ is not invertible. I have ...
-1
votes
4answers
56 views

If $A$ is a $3 \times 3$ matrix and $\det(A) = 4$, then compute $\det(((-9A)^4)^T)$. [closed]

Given a $3\times3$ matrix $A$ $$A= \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} $$ and $\det(A)=4$ Calculate $\det(((-9)\cdot A)^4)^T$.
1
vote
2answers
21 views

If rank of $(m+1)\times n$ matrix is $m+1$, then some $(m+1)\times (m+1)$ submatrix has non-zero determinant.

I can't understand this : If I have a $(m+1)\times n$ matrix and if its rank is $m+1$, then some $(m+1)\times (m+1)$ submatrix has non-zero determinant. How is it so?... kindly help.
4
votes
1answer
50 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 ...
2
votes
1answer
30 views

Writing the scalar product using a determinant

Let $A \in \mathbb{R}^{n \times n}$ be symmetrical and positive definite. Does the following statement hold true for $x \in \mathbb{R}^n$? $$\det(x^TAx) = \det(x^TxA)$$ And if so, how can it be ...
0
votes
0answers
23 views

Derive the determinant of circulant

Let $$ \sigma\in S_n $$ denote the permutation given by $$\sigma\in \begin{pmatrix} 1 & 2 & 3 & ...& n\\ n & 1 & 2 & ... & n-1\\ \end{pmatrix} $$ and let $$ P = ...
0
votes
1answer
18 views

xA=0 sufficient condition for zero determinant?

Let A be a symmetric n by n matrix and x be a 1 by n vector. If I find one x such that xA=0, does it mean A is singular?
0
votes
0answers
9 views

Maximizing determinant of addition of two matrices

Maximizing the determinant of addition of a constant matrix and a variable matrix is same as maximizing the variable matrix alone?
0
votes
1answer
42 views

If A is some invertible $n \times n$ matrix then show $\det(A^n) = (\det(A))^n$ for all $n\in \mathbb{Z}$

So there exists $A^{-1}$. I am assuming $\det(AB)=\det(A)\cdot\det(B)$ and $(A^d)^f=(A^{df})$ I know the proof for $\det(A^{-1})=(\det(A))^{-1}$ is: $\det(I_n)=1$ $\det(A\cdot A^{-1})=1$ ...
1
vote
2answers
51 views

Determinant matrix proof

Let $A$ be an $n\times n$ matrix and $i,j,k$ be $1\leq i,j,k\leq n$ and $\alpha,\beta \in \mathbb{R}$. I am supposing that $\bf{a}_k$(the $k$-th row) is equal to $\alpha \bf{a}_i+\beta \bf{a}_j$. ...
0
votes
0answers
35 views

Prove statement about cofactor.

Let $A$ be a $n$ x $n$ matrix $\in R$ and $det(A)=2$ , prove that atleast one of its cofactors is odd.