$a_{ij}$ is the entry in the ith row and jth column of A
diagonal entries are $a_{11}$, $a_{22}$, etc. and form the main diagonal. if non-diagonal entries are zero, then it’s a diagonal matrix.
equal matrices have same size and their corresponding entries are equal.
sum A+B: sum corresponding entries in A and B.
scalar multiple $rA$ is matrix whose columns are r times the corresponding columns in A (with r scalar).
the usual rules of algebra apply to sums and scalar multiples of matrices.
when matrix B multiplies vector x, it transforms x into vector $Bx$. if $Bx$ is multiplied by A, the result is $A(Bx)$. $A(Bx)$ is produced from x by composition of mappings, which can be represented as multiplication by a single matrix AB.
$A(Bx) = (AB)x$
$AB = A \begin{bmatrix} b_1 & b_2 & \dots & b_p \end{bmatrix} = \begin{bmatrix} Ab_1 & Ab_2 & \dots & Ab_p \end{bmatrix}$
A is matrix, B is matrix with columns $b_1 \dots b_p$.
each column of AB is a linear combination of columns of A using weights from corresponding column of B. AB has the same number of rows as A and same number of columns as B. if the number of columns of A does not match number of rows of B, the product is undefined. in general, AB ≠ BA.
if product AB is defined, then:
$(AB)_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j} + \dots + a_{in} b_{nj}$
$row_i (AB) = row_i (A) \times B$
a nifty trick: if multiplying matrix m×n by matrix n×o, the product will be a matrix m×o.
$A^k = \underbrace{A \dots A}_{k}$
with $A$ an n × n matrix and k a positive integer.
a matrix $A’$ whose columns are made up of the corresponding rows of $A$
properties:
the transpose of a product of matrices == product of their transposes in reverse order
invertible (singular) if there is same size matrix C such that $CA = I$ and $AC = I$ where I is the n × n identity matrix.
identity matrix: a matrix where the diagonals are all 1.
C is uniquely determined by A, so: $A^{-1} A = I$.
let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}.$ if $ad - bc \ne 0$ then $A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$
determinant: $\det A = ad - bc$
if A is invertible (determinant is not 0), then for each $b \in \Re^n$ the solution of $Ax = b$ is $A^{-1} b$.
properties of inverse:
finding $A^{-1}$:
elementary matrix: obtained by performing single elementary row operation on identity matrix
if elementary row operation is performed on m × n matrix A, result is EA, with E an m × m matrix obtained by performing same row operation on $I_m$
inverse of any elementary matrix E is of same type that transforms E back into I.
an n² matrix A is only invertible if A is row equivalent to $I_n$. any sequence of elementary operations reducing A to $I_n$ also transforms $I_n$ into $A^{-1}$.