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,
(1001) and (10 0001 00 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
(−2437).
(a)(1001)(b)(0110)
Solution:
(a)(−2437)(1001)=(−2×1+4×0−2×0+4×13×1+7×03×0+7×1)=(−2437)Therefore,(1001)is an identity matrix.(b)(−2437)(0110)=(−2×0+4×1−2×1+4×03×0+7×13×1+7×0)=(4−273)≠(−2437)Therefore,(0110)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)(−3257)(1001)and(1001)(−3257)(b)(0011)(1853)and(1853)(0011)
Solution:
(a)(−3257)(1001)=(−3×1+2×0−3×0+2×15×1+7×05×0+7×1)=(−3257)(1001)(−3257)=(1×−3+0×51×2+0×70×−3+1×50×2+1×7)=(−3257)∴(1001)is an identity matrix for(−3257).(b)(0011)(1853)=(0×1+0×50×8+0×31×1+1×51×8+1×3)=(00611)(1853)(0011)=(1×0+8×11×0+8×15×0+3×15×0+3×1)=(8833)∴(0011)is NOT an identity matrix for(1853).