4.6 Identity Matrix

4.6 Identity Matrix
1. Identity matrix is a square matrix, usually denoted by the letter I 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 I is the identity matrix of order n × n and A 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 ).