MAT 2401 Linear Algebra

MAT 2401Linear Algebra 2.3 The Inverse of a Matrix http://myhome.spu.edu/lauw

HW • WebAssign 2.3 • Written HW

Preview • The definition of the Inverse of a matrix. • Formula of the inverse for 2x2 matrices. • Use row operations to find the inverse of nxn matrices.

Recall

Recall Identity Matrix nxn Square Matrix

Inverse Matrix • Let A be a square matrix. Then the inverse for A is a square matrix A-1 of the same size as A such that AA-1 = I = A-1A

Inverse Matrix • Let A be a square matrix. Then the inverse for A is a square matrix A-1 of the same size as A such that AA-1 = I = A-1A • If such inverse A-1 exists, then the matrix A is said to be invertible (otherwise, singular).

Example 1 (a)

Inverse of a 2x2 Matrix

Example 1 (b)

Matrix Equations

Example 1 (c) Use matrix inverse to solve

Remarks • When n≥3, there are no useful formula to find the inverse of an invertible matrix.

How to Find A-1 ?

Example 1 (d) Use row operations to find the inverse of

Inverse of an 3x3 Matrix (Same for nxn matrices) Given matrix A, we set up the following matrix

Inverse of an 3x3 Matrix (Same for nxn matrices) Given matrix A, Use row operations to get to the second matrix. A-1(if exists) is the matrix on the right half.

Example 2 (a) Find the inverse of

Example 2 (b)

Q&A We can use two methods to solve a system of equations. (a) Gauss-Jordan Elimination (b) Matrix Inverse Q: Why use (b) when (a) is easier? A:

Properties of Matrix Inverse If A be an invertible matrix, kZ+ , c≠0 is a scalar, then A-1, Ak, cA, and AT are invertible. Also, 1. (A-1)-1= A 2. (Ak)-1 =(A-1)k 3. (cA)-1 = A-1 4. (AT)-1 =(A-1)T 5. (AB)-1 =B-1A-1

Cancellation Properties If C is invertible, then 1. AC=BC implies A=B 2. CA=CB implies A=B

MAT 2401 Linear Algebra