Matrix Inverses The Inverse of a Matrix Defn. The inverse of a - PowerPoint PPT Presentation

Matrix Inverses

The Inverse of a Matrix Defn. The inverse of a square matrix A , de- noted A − 1 , is the matrix such that AA − 1 = A − 1 A = I . Defn. The inverse is not guaranteed to exist. If it exists, then A is invertible ; otherwise A is not invertible or singular . invONE: 2

Matrix Equation with Invertible Matrix Fact. If matrix A is invertible, then A x = b has unique solution x = A − 1 b . invONE: 3

Inverse of a 2 × 2 Matrix The inverse of a 2 × 2 matrix has formula: � d − b � − 1 � a b � 1 = c d − c a ad − bc The formula also captures when the inverse ex- ists: the matrix is invertible if and only if ad − bc � = 0 . invONE: 4

Obtaining Matrix Inverses by Reduction One way to find the inverse is to solve the collec- tion of n vector equations A x = e 1 , . . . , A x = e n (where the e j are the columns of I n as before). Equivalently: To find inverse of matrix A , augment ALGOR with the identity matrix I n , then bring to re- duced row echelon form. invONE: 5

Example Inverse Calculation � 3 − 5 � 3 − 5 1 0 � � C = is augmented to − 5 9 − 5 9 0 1 � 1 0 9 / 2 5 / 2 � This reduces to 0 1 5 / 2 3 / 2 so that � 9 / 2 5 / 2 � C − 1 = 5 / 2 3 / 2 invONE: 6

Formulas Fact. If A and B are square matrices of the same size: (a) ( A − 1 ) − 1 = A (b) ( AB ) − 1 = B − 1 A − 1 (Note the reversal!) (c) ( A T ) − 1 = ( A − 1 ) T . invONE: 7

Characterization of Invertible Matrices The big theorem. For an n × n matrix A , the following are Fact. equivalent : ≻ A is invertible ≻ A has n pivots ≻ A is row equivalent to I n ≻ A x = 0 has a unique solution ≻ the columns of A are linearly independent ≻ the columns of A span R n ≻ the range of transform x �→ Ax is all of R n invONE: 8

Summary The inverse of a square matrix A is the matrix A − 1 such that their product is the identity. If inverse exists, then A is invertible; otherwise A is singular. If matrix A is invertible, then A x = b has unique solution A − 1 b . � � d − b The inverse of a 2 × 2 matrix has formula 1 ad − bc − c a One way to find the inverse is to augment with the identity matrix and bring to reduced row echelon form. invONE: 9

Summary (cont) An n × n matrix is invertible whenever it has n pivots; equivalently the columns are linearly in- dependent and span R n . invONE: 10