Question Bank - Mathematics

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Let A and B be non-singular matrices of the same order such that AB = A and BA = B. Which of the following statements is/are correct ?1. A2 = A2. AB2 = A2BSelect the correct answer using the code given below:

A.
1 only
B.
2 only
C.
Both 1 and 2
D.
Neither 1 nor 2

Solution:

Concept:Non Singular matrix is a square matrix whose determinant is a non-zero value. Calculation:Given:AB = A and BA = BStatement I: A2 = AWe are given that A = AB? A2 = (AB)2? A2 = A(BA)B? A2 = A(B)B [? BA = B]? A2 = (AB)B? A2 = (A)B [? AB= A]? A2 = A Statement I is true.Statement II: AB2 = A2BWe are given that B = BA? B2 = (BA)2? B2 = BABAA is pre-multiplied both sides, we get,? AB2 = ABABA? AB2 = ABA(BA)? AB2 = ABA(B)? AB2 = (AB)AB? AB2 = (A)AB? AB2 = (AA)B? AB2 = A2BStatement II is true.? Statement I and II both are true.

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