AX = I, so A and X have the same size

if AX = I, then XA = I

AX = I, then A and X are squared matrices

in 2x2 matrix if ad-bc = 0 The matrix can has an inverse

to get the inverse of A 2x2 matrix we can use this formula: 1/ad-bc .B where B is the same as A with swapping diagonal elements and multiply non-diagonal elements by -1, and to make sure that ad-bc not equal to 0

If A is invertible, the reduced row-echelon form of A is I

if A is invertible AX = b has exactly one solution for every nx1 vector b

if A is invertible AX = 0 has exactly one solution (namely trivial solution x=0)

if A is invertible then AX = b has exactly one solution namely A^-1 .b

if A isn’t invertible, then AX = b has either infinite solutions or no solutions

(A^-1)^-1 =

(5A)^-1 = 1/5 A^-1

(AB)^-1 =

if a product AB isn’t invertible then A or B isn’t invertible

if A or B isn’t invertible then AB isn’t invertible

ax + by = c          ( a ≠ 0 or b ≠ 0 ) is an example of linear equation in Three-dimensional

linear equation can involve products or roots of variables

all variables in linear equations occur only to the first power and don't appear

aij in a linear system, the ……. subscript indicates the equations in which the coefficient occurs and the …….indicates which unknowns it multiplies.

We say that a linear system is inconsistent If it has at least one solution

We say that a linear system is ……. If it has at least one solution and …….. if it has no solutions