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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2answers
81 views

What is the determinant of []? [closed]

I typed this in Matlab, but I can't understand why it returns the determinant one. A = [] det(A) ans = 1
1
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3answers
53 views

Prove that the product of two invertible matrices also invertible

I am working on a homework problem, but I am lacking some understanding. Here is the problem: Let $A$ and $B$ be invertible $n \times n$ matrices with $\det(A) = 3$ and $\det(B) = 4$. I know that ...
1
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2answers
44 views

Determinant of map $p(x) \mapsto (Tp)(x)=a_n+a_{n-1}x+ \ldots +a_0x^n$

Let $V$ be the vector space of polynomial $\mathbb{R}$ of degree less than or equal to $n$. For $p(x)=a_0+a_1x+ \ldots +a_nx^n$ in $V$. Define a Linear Transformation $T:V \to V$ by ...
0
votes
0answers
22 views

As a square matrix's size increases from dimension 2 to say 50, how does the variance of the matrix's determinant change? [duplicate]

both for the situation where elements in the matrix are randomly assigned any number and where elements are assigned a value uniformly distributed between 0 and 1. Thanks so much! (This is not a ...
1
vote
1answer
33 views

What is the determinant of cofactor matrix of a matrix? [duplicate]

For an $n \times n$ square matrix $A$, can determinant of its cofactor matrix (matrix consisting of cofactors of the elements of $A$) be expressed in terms of $\det(A)$ and $n$ ?
0
votes
1answer
83 views

Find determinant of given matrix

Let $A$ be an $n × n$ matrix of the following form. What is the value of the determinant of $A$? My attempt: I've used brute force to identity correct option. When I put $n=1$, then ...
1
vote
0answers
38 views

How do I know that $\det(a,b)$ is the area of parallelogram?

Please give an easy explanation, high school level.
0
votes
1answer
73 views

Easy way to get Determinant of 4 by 4 matrix

I have learned one way to get $4\times 4$ determinant. That is, divide a matrix $A$ by 4 part where each part is $2\times 2$ matrix: $$A = \left(\begin{array}{cc} B & C \\ D & E ...
7
votes
2answers
122 views

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 ...
0
votes
1answer
33 views

For what values of $x_1, x_2, x_3, x_4$ is the matrix $A$ invertible? [duplicate]

For what values of $x_1, x_2, x_3, x_4$ is the matrix $A=\begin{pmatrix}1 & 1 & 1 &1 \\ x_1 & x_2 & x_3 &x_4 \\ x_1^2& x_2^2 & x_3^2 & x_4^2\\ x_1^3& x_2^3 ...
3
votes
1answer
50 views

Determinant of $P_n$

I am preparing for an exam on linear algebra within few days, so I am in desperate need for a solution for the following question: Question: Let $P_n$, $n\ge2$, be the $n\times n$ matrix whose ...
0
votes
0answers
9 views

Determinants using Row Reduction replacement

I am aware replacement does not affect the value of determinant when doing a row reduction. However, I realised there isn't a good explanation on how to handle different forms of replacement when ...
0
votes
0answers
22 views

Find the value of $21D$ when elements of matrix are in H.P.

If $det$ represents and determinant and $$ \det\begin{bmatrix} a_1 & a_2 & a_3 \\ 5 & 4 & a_6 \\ a_7 & a_8 & a_9 \end{bmatrix}=D $$ and $a_1,a_2,a_3,5,4,a_6,a_7,a_8,a_9$ are ...
0
votes
0answers
34 views

What is relation between these two determinants?

Let $x\ne 0$ and $A$ be a square matrix of order $n$ and $$ B=\operatorname{diag}\begin{bmatrix} x & x & \ldots&x&x&x&x+\frac{2}{x} \end{bmatrix}$$ $$ ...
0
votes
2answers
14 views

Choose the correct option for the following determinant

Do we have to expand the determinant to find sum of Coefficients or coefficient of any power of $x$ or can it be calculated without expanding too?
0
votes
1answer
32 views

Different determinant for same matrix

I have the following matrix: $$ A=\begin{bmatrix} 2883,4675 & 44263,069125 & 724401,86824027 \\ 44263,069125 & 724401,86824027 & 12346864,4095603\\ 724401,86824027 & ...
0
votes
1answer
45 views

Various matrix manipulations effect on determinant

Suppose that the matrix $A$ below has determinant $−3$. Find the determinants of $B$, $C$ and $D$. $$ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} ...
3
votes
2answers
44 views

Solving generalized determinant related

How to solve following determinant by applying suitable elementary row/column transformations to obtain characteristic polynomial? \begin{align*} \left\vert \begin{matrix} -\lambda & 0 & 1 ...
6
votes
3answers
112 views

A is an antisymmetric matrix (of even size). B is another matrix such that $b_{i,j}=a_{i,j}+c$. Prove that |A|=|B|

I know that B would look something like this: $$\begin{bmatrix} c & a_{12}+c &...&&a_{1n}+c \\ -a_{12}+c & c &...&&a_{2n}+c \\ . \\ . \\ . \\ -a_{1n}+c & ...
5
votes
1answer
111 views

$\det(ABC) = \det(B)\det(AC)$?

Suppose $A$, $B$, and $C$ are $(n \times m)$, $(m \times m)$, and $(m \times n)$ matrices respectively, with $m\gt n$. What are the most general conditions under which $$ \det(ABC) = ...
1
vote
1answer
49 views

Block Matrix Determinant (Eigenvalues)

I have a matrix whose determinant is zero: $$\det\begin{bmatrix}A-I\lambda&B\\C &D-I\lambda \end{bmatrix} = 0$$ where $\lambda$ is a vector of complex scalars, I is an identity matrix, and ...
0
votes
1answer
33 views

General conditions for submatrices in regards to determinant

What are the most general conditions on sub-matrices A,B,C,D st det[A B;C D] = det(AD-BC) Obviously this is how determinant is defined for a regular square 2x2 matrix, but I don't understand how to ...
1
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1answer
23 views

Describing product of diagonal terms in Smith normal form in term of minors

Suppose $A$ is an $n\times n$ integer matrix. Then $A$ is equivalent to a matrix with diagonal entries $d_1,d_1,\dots,d_n,$ where $0<d_i\mid d_{i+1}$ for $1\leq i\leq n-1.$ I'm asked to describe ...
1
vote
1answer
21 views

$m$ order determinant related

How to find following $m$ order determinant? $\begin{vmatrix} 1&1&1&1&1&\cdots&1\\ 1&-1&0&0&0&\cdots&0\\ 1&0&-1&0&0&\cdots&0\\ ...
4
votes
1answer
231 views

Principal Minors of $B(AB)^{-1}A$ and Cauchy-Binet Terms

I am looking for a proof for the following conjecture. I think the result follows from applying a generalization of the Cauchy-Binet formula to the matrix $\mathbf{M}$ defined bellow. I've tested it ...
0
votes
3answers
38 views

determinant of SU(3) matrix

I don't understand the determinant condition on SU(3) group, broadly. I know that the determinant of such matrices should be equal to 1. But what is the real intention of that 1? Is it the real ...
-1
votes
1answer
10 views

linear algebra computation of area using determinant [closed]

Let $P = (1,0,-1)$ and $Q = (1, 1, 1)$ and $R = (2,2,1)$. Choose $S$ so that $PQRS$ is a parallelogram and compute its area. Choose $T$, $U$, $V$ so that $OPQRSTUV$ is a tilted box and compute its ...
2
votes
0answers
23 views

Principal Minors of $B[AB]^{-1}A$

Suppose $A$ and $B$ are $(n \times m)$ and $(m \times n)$ matrices respectively, with $n<m$ and $\operatorname{rank}(A)=\operatorname{rank}(B)=n$. Consider the matrix $M$ given by $$ M = ...
1
vote
3answers
98 views

the minimum and maximum values of the determinant of order $3\times3$ matrix with entries $\{0,1,2,3\}$

The minimum and maximum values of the determinant of order $3\times3$ matrix with entries $\{0,1,2,3\}$.
0
votes
2answers
73 views

calculation of determinant: why a(a b c d) will be a^{2}?

I've just started to study linear algebra, and this is my 1st post to this site. the problem I understand that if $\mathbf {X=[a\quad b\quad c\quad d]}$ then the determinant will be $\mathbf {aX+b ...
0
votes
0answers
29 views

Finding the determinant of this general case matrix

I'm getting some headache when I want to find the result of this general matrix matrix. Things works ok when I have one with numbers, but when it comes to this kind of matrix I'm totally lost. I use ...
5
votes
1answer
100 views

Prove that the determinant of an invertible matrix $A$ is equal to $±1$ when all of the entries of $A$ and $A^{−1}$ are integers.

Prove that the determinant of an invertible matrix $A$ is equal to $±1$ when all of the entries of $A$ and $A^{−1}$ are integers. I can explain the answer but would like help translating it into ...
6
votes
4answers
473 views

What is the intuitive meaning of a determinant? [duplicate]

I know how to calculate a determinant, but I wanted to know what the meaning of a determinant is? So how could I explain to a child, what a determinant actually is. Could I think of it as a measure ...
0
votes
1answer
39 views

Determinant manipulation involving variable

This manipulation was done by somebody else and I am trying to understand what is happening: $$\det \begin{bmatrix} \lambda-1 & 1 & 1 \\ 1 & \lambda - 1 & 1 \\ 1 & 1 & \lambda ...
1
vote
1answer
25 views

Non-Changing Determinant When Adding (Seemingly) Arbitrary Entries

Question: I've found that adding what seem to be arbitrary values in the 4th row don't change the value of the determinant. Why is that? A = $\begin{bmatrix} 0 & 2 & 0 & 0 & 0 \\ 0 ...
0
votes
2answers
24 views

Finding the determinant of this matrix B, using the determinant of Matrix A.

We are provided with a matrix and it's corresponding determine: This Matrix, let's call it A, has a determinant of 4. \begin{bmatrix} a & b & c \\ d & e & f \\ ...
-1
votes
1answer
48 views

Determinant of special tridiagonal matrix

I want to compute determinat of the following tridiagonal matrix: $$A = \begin{pmatrix} 1 & -1 & 0 & \cdots & \cdots & 0 \\ 1 & 1 & -1 & 0 & \cdots & 0 \\ 0 ...
-3
votes
1answer
25 views

Prove that an mxn matrix with m<n has no left inverse, similarly an mxn martix with m>n has no right inverse

I wonder how can I prove A matrix $A_(mxn)$ with $m \lt n$ has no left inverse and a matrix $A_(mxn)$ with $m \gt n$ has no right inverse Because I got no idea about that
1
vote
2answers
40 views

How to determine is a matrix is invertible?

$$ M= \left[ \begin{array} -1 & k & 3 & -2\\ 2 & 1 & 1 & k \\ 1 & k+1 & 4 & k-2\\ 2 & 1 & 1 & k+1\\ ...
2
votes
1answer
22 views

Determinant property

My question is: "If determinant of $A^2$ is $0$ can we also say that determinant of $A$ is $0$?" I have tried to argue that it is by saying that if $A^2 = 0$ then $A$ must be $0$ so determinant is ...
0
votes
1answer
16 views

Suppose that $A \in M_{5\times5}$, and suppose that $RREF_A = B$. Find $\text{rank}(A)$, if we know that $\det A = \det B + 1$

Suppose that $A \in M_{5\times5}$, and suppose that $RREF_A = B$. Find $\text{rank}(A)$, if we know that $\det A = \det B + 1$ I'm not really sure how to approach this question. I know that $n = 5$ ...
1
vote
0answers
36 views

Alternate formula for the determinant of a 3x3 Matrix

So, I've recently been reading about Dodgson condensation, for matrices. Take, in particular the $3x3$ case. Define a matrix, $|A|:$ \begin{pmatrix} a & b & c \\ d ...
3
votes
1answer
79 views

What is an intuition behind permanent?

I would like to know what is your intuition behind permanent of a matrix. For me, it looks like someone came and saw determinant, deleted permutation sign and voila, we have permanent and it counts ...
2
votes
2answers
65 views

Determinant of $N \times\ N$ matrix

So the question asks: For $n \geq 2$, compute the determinant of the following matrix: $$ B = \begin{bmatrix} -X & 1 & 0 & \cdots & 0 & 0 \\ ...
1
vote
1answer
38 views

Prove the theorem of row swapping determinants??

I don't know how to prove this theorem in a clear way, I could really use some help, Thanks so much! This is a linear-algebra problem dealing with determinants. Let M' be the matrix obtained from ...
2
votes
2answers
40 views

In which situations $\det(A\mod x) \mod x=\det(A)\mod x$ would help us knowing if $\det(A)=0$?

André Nicolas, in his very neat answer to is the following matrix invertible? uses the fact that the matrix $$ \begin{bmatrix} 1235 &2344 &1234 &1990\\ 2124 & 4123& 1990& 3026 ...
0
votes
0answers
28 views

Finding Factors of a Determinant

Consider the determinant with elements: $a_{11} = ax-by-cz, a_{12}=ay+cz, a_{13}=cx+az$ $a_{21}=ay+bx, a_{22}=by-cz-ax, a_{23}=bz+cy$ $a_{31}=cx+az, a_{32}=bz+cy, a_{33}=cz-ax-by$ Where $a_{ij}$ ...
0
votes
1answer
18 views

Jacobian of a diffeomorphism

Let $U,V\subseteq \mathbb{R}^{n}$ be open. Let $\alpha:U \to V$ be a smooth homeomorphism. Furthermore, assume that $\mathcal{J}_{\alpha}(\mathbf{x})$ (the Jacobian matrix) has rank $n$ for all ...
2
votes
0answers
48 views

A brief answer for the determinant of a matrix

I am given the following matrix $A=(a_{ij})_{6 \times 6}$, where $a_{ij}=\sum_{k=1}^{10} x_k^{i+j-2}$. Remark: If $A=(a_{ij})_{10 \times 10}$ with the same $a_{ij}$ defined above, the answer is very ...
1
vote
1answer
58 views

What is the average determinant of a matrix in $M_{2}(\mathbb{Z}/n\mathbb{Z})$?

Given a finite collection of matrices, it is natural to consider the average determinant of a matrix in such a collection. For example, an interesting problem involving the average determinant of a ...