**3.694 determinant questions.**

Prove that if $A$ has linearly dependent rows or columns then $\det(A)=0$.
$$\DeclareMathOperator{\sgn}{sgn}
\det(A)=\sum_{\sigma\in s_n}\Bigl(\sgn(\sigma)\prod_{i=1}^{n}a_{i,\sigma(i)}\Bigr)=\sum_{\...

I have started studying linear algebra and I came across determinants. I know there are all sorts of formulae for computing it but I can't find what exactly it is. Why are they calculated the way ...

$A$ is an $n\times n$ matrix and $L_A$ is the left-multiplication operator on the $n\times n$ matrices.
I have seen this question and the given answer. But I could not understand the answer. I have ...

Let me first describe where I start:
$$\iint_Sz^2\,dS$$
$dS$ is the octant of a sphere.
The radius = 1.
The sphere is centered at the origin.
$R$ is the projection of $S$ on the $xy$-plane.
$$S=x^...

The first derivative of the determinant function is well-known and is given by Jacobi formula: let $A(t)$ be a matrix function of scalar variable $t$, then
$$
d~\text{det}(A(t))=\text{tr}\Big(\text{...

I have a question regarding the following determinant:
$\begin{vmatrix}
+ax - by - cz & bx+ay & cx+az \\
bx+ay& -ax+by-cz & bz+cy \\
cx+az & bz+cy & -ax-by+cz
\end{vmatrix}
=
...

How can I prove that if a linear map $T \ \in \ \text{End}(V)$ with $\dim V < \infty $ has eigenvalues ${\lambda}_{1},{\lambda}_{2}, ..., {\lambda}_{n} $, then $\text{trace}(T)$ $=$ ${\lambda}_{1}+...

I'm currently taking multivariable calculus, and I'm familiar with the second partial derivative test. That is, the formula $D(a, b) = f_{xx}(a,b)f_{yy}(a, b) - (f_{xy}(a, b))^2$ to determine the ...

Check If arguments are correct to their corresponding questions :
Q question 1
Argument1. Options A, B , D are incorrect beacuse idempotent is only valid in multiplication ( as what definition states ...

Knowing that $$\begin{vmatrix}
2 & 2 & 3\\
x & y & z\\
a & 2b & 3c
\end{vmatrix}=10$$ and $x,y,z,a,b,c \in \mathbb{R}$, calculate
$$\begin{vmatrix}
0 & 3x & y & z\...

I've tried several ways of solving this but I can't get the answer correct.
the answer is a=-2 and a=-4

I can solve for the cases in which $C=0$ or $B=0$. But I cannot find any idea for the case in which both $B$ and $C$ are nonzero matrices.

I know the answer is $$\det M_B = |\det B|^{2n}.$$ I am trying to prove it by considering this operator as a composition of two linear operators right multiplication and left multiplication. Am I ...

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) - \...

We know that if $A$ is a matrix with rows $A_1, ..., A_n$, then $\sigma(A)$ is a matrix with rows $A_{\sigma(1)}, ..., A_{\sigma(n)}$.
Actually, one can show that $\sigma(A) = \sigma(I)A$, where $I$ ...

In given below picture
How can we interchange the A position to between A^k and B^k+1 . As given condition is only stating AB=BA , NOTHING ABOUT POWERS

Why that highlighted condition is true ? How can we prove it ? I have no idea about elementary operation in this specific way i.e matrix equation and also in row operation we have to do something ...

Evaluate $$D=\begin{vmatrix}
-2a &a+b &a+c \\
b+a& -2b &b+c \\
c+a&c+b & -2c
\end{vmatrix}$$
My try:
Applying $R_1 \to R_1+R_2$ we get
$$D=\begin{vmatrix}
b-a&a-b ...

I am given the determinant of matrix $A$ and matrix $B$. Both are $3 \times 3$ matrices.
$\det(A) = -5$
$\det(B) = 5$
I need to find the determinant of $D = 6A^{-1}B^T$.
I calculated the ...

Let $A=(a_{ij}) \in M_{n}(\mathbb{R})$, $a_{ii} >0$ $~$ for $i$ from $1$ to $n$ and $a_{ij} \leq 0$ for $i\neq j$ ,$1\leq j\leq n$, $\sum_{i=1}^{n}a_{ij} >0$.
Prove that $detA>0$

I am trying to find an orthogonal matrix $P$ of the quadric $3x^2+3z^2-4xy-8xz+4yz=1$.
I have found that the eigenvalues are $-1$ (with algebraic multiplicity 2) and $8$ (with algebraic multiplicity ...

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.

Show that
$$\det
\left(\begin{array}{cc}
0 & A\\
-B & I
\end{array}\right) = \det(AB)
$$
where A, B are compatible matrices, 0 and I are zero and identity matrices of the appropriate size.
...

From some abstract considerations I know that $$\det \pmatrix{A & -\bar{B} \\ B & \bar{A}}= \det \pmatrix{\bar{A} & \bar{B} \\ -B & A}$$ for $A,B$ complex square matrices satisfying $A^...

A symmetric matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$.
However, such matrices can also be characterized by the positivity of the principal minors.
A statement and proof can, ...

Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then
$$ \det(I_m+AB) = \det(I_n+BA)$$
where $I_m$ and $I_n$ denote the $m \times m$ ...

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

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 $(...

Let $n\in \Bbb N$ and $\mathcal{M}_n(\Bbb R)$ be the set of the square real-valued matrices.
Find all $A \in \mathcal{M}_n(\Bbb R)$ such as $|\det(A)| = \prod_{i=1}^n\left( \sum_{j=1}^n |a_{i,j}| \...

Given a $n\times n$ P.D. matrix $A$ and a diagonal matrix $D = \mathrm{diag}(d_1,\ldots,d_n)$, with $d_i > 0$, $i=1\ldots,n$, I want to prove that the product $AD$ is also positive definite.
Since ...

i have to prove:
$$\begin{vmatrix}
-2a &a+b &a+c \\
b+a&-2b &b+c \\
c+a&c+b &-2c
\end{vmatrix} = 4(a+b)(b+c)(c+a)$$
I have tried many calculations between the rows ...

Calculate the determinant of the matrix $$A=\begin{pmatrix} \sin\alpha
& \cos\alpha & a\sin\alpha & b\cos\alpha & ab \\
-\cos\alpha & \sin\alpha & -a^2\...

When calculating the area of a triangle (using determinants). What is the meaning of a negative determinant? Is it ok to just turn the value into a positive one?

Prove that the set of $n \times n$ matrices with determinant $1$ is unbounded closed with empty interior in $\mathbb{R}^{n^{2}}$.
The aplication $\det$ is continuous, so the inverse image of a closed ...

Let $D\in\mathbb{R}^{n\times n}$ be a real diagonal matrix where $\sum_i D_{ii}<0$. Let also $R\in\mathbb{R}^{n\times n}$ and $L\in\mathbb{R}^{n\times n}$ be real (possibly) non-symmetric (...

Let $A$ be a $n \times n$ invertible matrix, prove $\det(kA)=k^n\det A$. I really don't know where to start. Can someone give me a hint for this proof?

Solve Question D3 of the image whose link is given below
https://i.stack.imgur.com/xS576.jpg
My approach:
I made a matrix of all the coefficients say A (3*3) and then I made a column matrix of ...

If $A$,$B$,$C$ are Square matrices satisfying $$(A-B)C=BA^{-1},$$ where $A$ is nonsingular.
Then which is true among these?
$C(A-B)=BA^{-1}$
$(A-B)C=A^{-1}B$
$C(A-B)=A^{-1}B$
$(A-B)^{-1}=C+A^{-1}$
...

The problem is to find the second order term in the series expansion of the expression $\mathrm{det}( I + \epsilon A)$ as a power series in $\epsilon$ for a diagonalizable matrix $A$. Formally, we ...

Could you help me with this task? I should decide which of the two following statements in (i) and (ii) are true. I would appreciate it, if you would explain to me the solution in detail, because I ...

If $\lambda$ is an eigen value of an orthogonal matrix $A$, then show that $\frac{1}{\lambda}$ is also an eigen value of $A$, with the same set of eigen vectors.
I have proceeded like this:
If $A$ is ...

Justify that the determinant of the following matrix is zero
$$A=\begin{pmatrix} 2 & 1 & 0 & 5\\
-1 & 1 & 1 & 6\\ 5 & 1 & -1 & 4\\ 5 & 1 & 3 &...

Calculate the determinant of this magic square (which is from Albrecht
DÃ¼rer's Melancholia)
$$\begin{pmatrix} 16 & 3 & 2 & 13 \\ 5 & 10 & 11 & 8
\\ ...

For $2 \times 2$ matrices $A = \begin{pmatrix} a_{11} & a_{12}\\ a_{21} &
a_{22} \end{pmatrix}$ prove that: If row-vectors of $A$ are linearly
dependent, then $\det(A)=0$
I'm not sure how ...

I had a doubt, can we apply the determinant of a 2x2 Matrix to:
$$\begin{vmatrix}
E_{m} & A \\
B & E_{n} \\
\end{vmatrix} $$ with $A \in K^{m\times n}$ and $B \in K^{n\times m} $ and $E$ as ...

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix. I have already asked a (viewed but unanswered) ...

I'm trying to prove a formula I have constructed for the determinant of a general $n\times n $ real matrix $A$, given here in the case $n=5$:
$$ A = \begin{bmatrix}
1 & 1 & 1 & 1 & 0 \\...

I have an equation of the form:
$$A(s,t)x = b(s)$$
where $A$ is a real matrix, $x, b$ vectors and $s, t$ scalar parameters.
While generally speaking, if $A$ is singular, the equation generically ...

I wish to find the area of a parallelogram formed by two vectors in three dimensions.
Computing this is simple, taking the absolute value (modulus) of the cross product of the vectors, where the ...

If I am given a set of points in $\mathbb{R}^3$ and asked if they are collinear or not, can I test the set of points for linear independence to see if they lie on the same line or not? Then I would ...

- linear-algebra
- matrices
- eigenvalues-eigenvectors
- inverse
- polynomials
- trace
- abstract-algebra
- vector-spaces
- matrix-equations
- proof-verification
- matrix-rank
- systems-of-equations
- geometry
- inequality
- linear-transformations
- combinatorics
- derivatives
- matrix-calculus
- permutations
- algebra-precalculus
- multivariable-calculus
- real-analysis
- calculus
- differential-equations
- vectors

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