1.5 Elementary Matrices and a Method for Finding Inverse of A

Inverse Row Operations

If matrix B results from matrix A by performing elementary row operations, then the matrix A can be recovered from B by performing the corresponding inverse operation:

1. Multiply the same row by 1/c

2. Interchange the same two rows

3. If B resulted by adding c times row r_i of A to row r_j, then add −c times r_j to r_i

Row Equivalent

Matrices A and B are said to be row equivalent if either (hence each) can be obtained from the other by a sequence of elementary row operations.

Elementary Matrix

A matrix E is called an elementary matrix if it can be obtained from an identity matrix by performing a single elementary row operation.

Theorem 1.5.1 (Row Operations by Matrix Multiplication)

If the elementary matrix E results from performing a certain row operation on I_m and if A is an m×n matrix, then the product EA is the matrix that results when this same row operation is performed on A.

Theorem 1.5.2 (Inverse of Elementary Matrices)

Every elementary matrix is invertible, and the inverse is also an elementary matrix.

E₀E = I and EE₀ = I, let E₀ be the inverse row operation performed on I.

Theorem 1.5.3 (Equivalent Statements)

If A is an n×n matrix, then the following statements are equivalent, that is, all true or all false:

(a) A is invertible

(b) Ax = 0 has only the trivial solution

(c) The reduced row echelon form of A is I_n

(d) A is expressible as a product of elementary matrices

Row Operations to Find A⁻¹

To find inverse of matrix A, reduce A to identity matrix using row operations while simultaneously applying the same operations to I to produce A^-1

[ A | I ] → [ I | A^-1 ]

If zero row or column is made in the process, A is not invertible.