Determinants, eigenvalues and eigenvectors

If M is an n x n matrix with n>1, what is the determinant of M?

$$\det M=\sum_{k=1}^{n}\left(-1\right)^{k+1}M_{1k}\det\overline{M}_{1k}$$

If M is an n x n matrix with $1\le i\le n\land1\le j\le n$, then what does Laplace’s Expansion Theorem say?

-$detM=\sum_{k=1}^{n}\left(-1\right)^{i+k}M_{ik}\det\overline{M}_{ik}$ (expanding along ith row)

-$detM=\sum_{k=1}^{n}\left(-1\right)^{k+j}M_{kj}\det\overline{M}_{kj}$ (expanding along jth column)

Let M be an n x n upper triangular matrix over a field, then what is the detM equal to?

$\Pi_{i=1}^{n}Mii$ (product of entries on main diagonal)

Let A and B be n x n matrices over a field. If $A\overrightarrow{r_{i}\rightarrow\alpha r_{i}}B$ , then what is the detB?

$$\alpha\det A$$

Let A and B be n x n matrices over a field. If $A\overrightarrow{r_{i}\rightarrow r_{i}+\lambda r_{j}}B$, then what is the detB?

detA

Let A and B be n x n matrices over a field. If $A\overrightarrow{r_{i}\leftrightarrow r_{j}}B$, then what is the detB?

-detA

What is the$det\left(E_{r_{i}\rightarrow\alpha r_{i}}\right)$?

$$\alpha$$

What is the$det\left(E_{r_{i}\rightarrow r_{i}+\lambda r_{j}}\right)$?

1

What is the$det\left(E_{r_{i}\leftrightarrow r_{j}}\right)$ ?

-1

Let A be an n x n matrix, then what is the adjugate of A?

$$B_{ij}=\left(-1\right)^{i+j}\det\overline{A}_{ji}$$

Let A be an n x n matrix and B be the adjugate of A, with $det\ne0$, then what is $A^{-1}$ equal to?

$$\frac{1}{\det A}B$$

Let PA = LU, then what is the detA?

$$det(A)=\left(-1\right)^{k}\Pi_{i=1}^{n}U_{ii}$$

Let V be a vector space over K and $T:V\rightarrow V$ be a linear transformation. When is a vector $v\in V$ an eigenvector of T with eigenvalue $\lambda\in K$?

if $v\ne0\land T\left(v\right)=\lambda v$

Let $T:V\rightarrow V$ be a linear map and suppose that $\lambda\in K$ is an eigenvalue of T, then what is the $\lambda$-eigenspace of T?

the subspace $E_{\lambda}=\left\lbrace v\in V\vert T\left(v\right)=\lambda v\right\rbrace$ (all eigenvectors with eigenvalue $\lambda$ and the zero vector}

What is the geometric multiplicity of an eigenvalue $\lambda$ of T?

the dimension of the corresponding $\lambda$-eigenspace ($E_{\lambda}$)

Let $T:V\rightarrow V$ be a linear map. If there are eigenvectors of T with distinct eigenvalues then what can we say about the set of eigenvectors?

it is linearly independent

What does it mean when a field K is algebraically closed?

if every polynomial $p\left(x\right)=a_0+\cdots+a_{n}x^{n},a_{i}\in K\land a_{n}\ne0$ can be factorised in the form $p\left(x\right)=b\left(x-c_1\right)\ldots\left(x-c_{n}\right),b,c_{i}\in K$

Which of ℝ and ℂ is algebraically closed?

ℂ

Let $A\in M_{n}\left(K\right)$ where K is an algebraically closed field and suppose$\lambda\in K$ is an eigenvalue of A, then what is the algebraic multiplicity of $\lambda$ as an eigenvalue of A?

the power K to which $\left(x-\lambda\right)$ appears in the factoriation of $p_{A}\left(x\right)$

Let $A\in M_{n}\left(K\right)$, then what is the trace of A?

$$tr\left(A\right)=A_{11}+\cdots+A_{nn}=\sum_{i=1}^{n}A_{ii}$$

Let K be an algebraically closed field and $\lambda_1,\ldots,\lambda_{n}$ be all the eigenvalues of $A\in M_{n}\left(K\right)$ then what is tr(A) and det(A)?

$$tr\left(A_{}\right)=\lambda_1+\cdots+\lambda_{n}$$

$$det\left(A\right)=\lambda_1\ldots\lambda_{n}$$