matrices

definitions

a matrix is singular if it is not invertible.

minor

the minor $M_{ij}$ is the determinant of the submatrix obtained by removing row $i$ and column $j$ from $A$.

$$M_{ij}=\det (A_{ij})$$

cofactor

the cofactor $C_{ij}$ is

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}.$$

determinant

if

$$\det (A)=0,$$

then the rows or columns are linearly dependent.

laplace expansion

the laplace expansion or cofactor expansion along row $i$ is

$$\det (A)=\sum _{j=1}^n{a}_{ij}{C}_{ij}.$$

you can also expand along a column.

inverse

the inverse is only defined for square matrices.

for an $n\times n$ matrix, the following are equivalent:

• $A^{-1}$ exists
• $\det (A)\ne 0$
• $\mathrm{rank}(A)=n$
• the columns and rows are linearly independent
• $Ax=0$ implies $x=0$
• $Ax=b$ has exactly one solution for every $b$

row and column operations

you can replace a row $R_i$ with a linear combination of $R_i$ and other rows, as long as the coefficient of $R_i$ is not $0$.

row swaps are allowed too.

echelon form

in echelon form, all entries below each pivot are $0$.

echelon means staircase.

pivots

a pivot is the first nonzero entry in a row of echelon form.

rank

the rank is the number of pivots.

if rank is $n$, then the columns are independent, so

$$Ax=0$$

has only the trivial solution $x=0$.

if some $x_i\ne 0$ gave

$$x_1{C}_1+x_2{C}_2+\cdots +x_n{C}_n=0,$$

then one column would be a linear combination of the others, so the columns would be linearly dependent.

eigenvalues

an eigenvalue is a scalar $\lambda$ such that

$$Av=\lambda v.$$

the vector changes only in scale, not direction, after $A$ is applied.

to find eigenvalues, solve

$$\det (A-\lambda I)=0.$$

eigenvectors

an eigenvector $v$ satisfies

$$(A-\lambda I)v=0.$$

$v$ is the direction, and $\lambda$ is the scale factor.