4.6 Identity Matrix

4.6 Identity Matrix
1. Identity matrix is a square matrix, usually denoted by the letter and is also known as unit matrix.
 
2. All the diagonal elements (from top left to bottom right) of an identity matrix are 1 and the rest of the elements are 0.
For example,
( 1 0 0 1 )  and  ( 1 0       0 0 0 1       0 0      1 )  are identity matrices .

3. If is the identity matrix of order n × n and is a matrix of the same order, then IA = A and AI = A


Example 1:
Determine whether each of the following is an identity matrix of ( 2 4 3 7 ) .
( a ) ( 1 0 0 1 ) ( b ) ( 0 1 1 0 )
 
Solution:
( a ) ( 2 4 3 7 ) ( 1 0 0 1 ) = ( 2 × 1 + 4 × 0 2 × 0 + 4 × 1 3 × 1 + 7 × 0 3 × 0 + 7 × 1 ) = ( 2 4 3 7 ) Therefore, ( 1 0 0 1 ) is an identity matrix . ( b ) ( 2 4 3 7 ) ( 0 1 1 0 ) = ( 2 × 0 + 4 × 1 2 × 1 + 4 × 0 3 × 0 + 7 × 1 3 × 1 + 7 × 0 ) = ( 4 2 7 3 ) ( 2 4 3 7 ) Therefore, ( 0 1 1 0 ) is not an identity matrix .  


Example 2:
Find the product of the following pairs of matrices and determine whether the given matrix is an identity matrix.

( a ) ( 3 2 5 7 ) ( 1 0 0 1 ) and ( 1 0 0 1 ) ( 3 2 5 7 ) ( b ) ( 0 0 1 1 ) ( 1 8 5 3 ) and ( 1 8 5 3 ) ( 0 0 1 1 )  
 
Solution:
( a ) ( 3 2 5 7 ) ( 1 0 0 1 ) = ( 3 × 1 + 2 × 0 3 × 0 + 2 × 1 5 × 1 + 7 × 0 5 × 0 + 7 × 1 ) = ( 3 2 5 7 ) ( 1 0 0 1 ) ( 3 2 5 7 ) = ( 1 × 3 + 0 × 5 1 × 2 + 0 × 7 0 × 3 + 1 × 5 0 × 2 + 1 × 7 ) = ( 3 2 5 7 ) ( 1 0 0 1 ) is an identity matrix for ( 3 2 5 7 ) . ( b ) ( 0 0 1 1 ) ( 1 8 5 3 ) = ( 0 × 1 + 0 × 5 0 × 8 + 0 × 3 1 × 1 + 1 × 5 1 × 8 + 1 × 3 ) = ( 0 0 6 11 ) ( 1 8 5 3 ) ( 0 0 1 1 ) = ( 1 × 0 + 8 × 1 1 × 0 + 8 × 1 5 × 0 + 3 × 1 5 × 0 + 3 × 1 ) = ( 8 8 3 3 ) ( 0 0 1 1 ) is NOT an identity matrix for ( 1 8 5 3 ) .

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