Matrix Algebra

Elementary Matrices

Construction of Elementary Matrix. Let be the $n\times n$ identity matrix. Then a new matrix is obtained by applying either of the row operations to :

Add a multiple of the -th row of by to the -th row of ;
interchange the -th row and the -th row of ;
multiply the -th row of by where $c \neq 0$ .

And the resulting matrix

is called an elementary matrix.

Properties of Elementary Matrices. Given an $n\times m$ matrix , the matrix multiplication results in performing the corresponding basic row operation on . Furthermore, the inverse matrix $E^{-1}$ of the elementary matrix is also the elementary matrix obtained respectively by

adding a multiple of the -th row of by to the -th row of ;
interchanging the -th row and the -th row of (thus, producing the same elementary matrix);
multiplying the -th row of by $c^{-1}$ .

EXAMPLE 7. Construct the $3\times 3$ elementary matrix corresponding to each of the following row operations:

Add a multiple of the 1st row by (-4) to the 3rd row.
Interchange the 1st and 2nd row.
multiplying the 3rd row by 5.

Then find the inverse for each elementary matrix.

Finding the Inverse Matrix. Let be a square matrix. Suppose that $A \sim I_n$ . Then we have a series of basic row operations which reduces to . Or, equivalently we have a series of elementary matrices $E_1, E_2, \ldots, E_p$ so that $E_p \cdots E_2 E_1 A = I_n$ . Comparing

$\displaystyle (E_p \cdots E_2 E_1) A = I_n$ and $\displaystyle \quad A^{-1} A = I_n,$

we have $A^{-1} = E_p \cdots E_2 E_1$ .