Section 2.2: The Inverse of a Matrix

Elementary Matrices

A matrix obtained by performing a single elementary row operation on an identity matrix. If an elementary row operation is performed on an m x n matrix A, the resulting matrix can be written as EA, where the m x m matrix E is created by performing the same row operation on the identity matrix of size m.

Nonsingular Matrix

A matrix that is invertible

Singular Matrix

A matrix that is not invertible

Invertible Matrix Theorem 2

An n x n matrix A is invertible if and only if A is row equivalent to the identity matrix and any sequence of elementary row operations that reduces A to I also transforms I into inv(A) i.e. Find the inverse by setting up [A | I] and row reduce until [I | inv(A)]

Invertible Matrix

An nxn matrix A is said to be invertible if there is an nxn matrix C such that CA = I and AC = I. In this case, C is the inverse of A.

Invertible Matrix Theorem 1

If A is an invertible n x n matrix, then for each b in R^n, the equation Ax = b has the unique solution x = inv(A) * b

Formula for Invertible Matrix

Let A = | a b | ... if ad - bc does not = 0, then: | c d | inv(A) = (1 / (ad - bc)) * | d -b | | -c a |

Invertible Matrix Facts

a. If A is an invertible matrix, then inv(A) is invertible and inv(inv(A)) = A b. If A and B are n x n invertible matrices, then so is AB, and the inverse of A is the product of the inverses of A and B in reverse order: inv(AB) = inv(B) * inv(A) c. If A is an invertible matrix, then so is trans(A), and the inverse of trans(A) is the transpose of inv(A): inv(trans(A)) = trans(inv(A))

Determinant

detA = ad - bc. A matrix is only invertible if and only if detA does not equal 0.

Property of Invertible Matrix

inv(A) * A = I and A * inv(A) = I