invertible matrix theorem!

14 guidelines of the invertible matrix theorem.

The Definition of Inverse

1) A is an invertible matrix; 2) There is a matrix where AX = I (identity matrix); 3) There is a matrix where XA = I (identity matrix)

The Property of Transposes

4) A^T is invertible (T → transpose; means to switch the rows and columns of a matrix)

The Standard Inverse Algorithm

5) A can be row reduced to I (identity matrix); 6) A has a pivot in every row & column

A Pivot In Every Column

7) Columns of A are linearly independent; 8) AX = 0 has a unique solution / no free variables; 9) T(x) = A(x) is one-to-one

A Pivot In Every Row

10) Columns of A span ℝ^n; 11) Ax = b (always consistent; have one unique solution or infinitely many solutions); 12) T(x) = A(x) is onto

Basis

13) Columns of A are a basis for ℝ^n

Determinant

14) A is invertible if det(A) = 0

How to Check for a Basis

7) Columns of A are linearly independent & 10) Columns of A span ℝ^n

OR simply 14) A is invertible if det(A) = 0