Question

Let A and B be two 2 × 2 matrices. Consider the statements
          (i) AB = 0 ⇒ A = 0 or B = 0
         (ii) AB = I ⇒ A = \[{B^{ - 1}}\]
          (iii) \[{\left( {A + B} \right)^2} = {A^2} + 2AB + {B^2}\]
A. (i) is false, (ii) and (iii) are true
B. (i) and (iii) are false, (ii) is true
C. (i) and (ii) are false, (iii) is true
D. (ii) and (iii) are false, (i) is true

Answer 1

Hint: Here, matrices A and B can be verified by taking examples of A and B. Randomly take A and B and check whether conditions are satisfied or not. Use basic concept of inverse matrix and basic properties of matrices.

Complete step-by-step answer:
Statement (i)
If A =$\left[ {\begin{array}{*{20}{c}}
  1&1 \\
  0&0
\end{array}} \right]$ and B = $\left[ {\begin{array}{*{20}{c}}
  0&1 \\
  0&{ - 1}
\end{array}} \right]$, then

 AB = $\left[ {\begin{array}{*{20}{c}}
  0&0 \\
  0&0
\end{array}} \right]$ = 0

Thus, AB = 0 but neither A = 0 nor B = 0.
So, statement (i) is false.

(ii) If A is a non-singular square matrix of order n then there exist a square matrix B of order n such that AB = BA = \[{I_n}\]
Then, \[{A^{-1}}\] = B.
Always choose an example method to verify these types of questions, if the order of the matrix is not so large i.e. 2 × 2 or 3 × 3. But if the order of the matrix is large then it will be difficult to verify these types of statements.
The statement Is true as the product AB is an identity matrix i.e., I, if and only B is inverse of the matrix A or A is inverse of B.

(iii) is false since matrix multiplication is not commutative.
\[{\left( {A + B} \right)^2} = {A^2} + AB + BA + {B^2}\]
In matrix $AB \ne BA$
Therefore, \[{\left( {A + B} \right)^2} \ne {A^2} + 2AB + {B^2}\]

So, the correct answer is “Option B”.

Note: In these types of questions, we can use basic definitions, concepts and properties to check whether the given statements are satisfied or not. Matrix system does not necessarily follow all rules of algebra.
Always remember that the inverse of a matrix exists for square matrices only.
Commutative law: If A = ${a_{ij}}$ and B = ${b_{ij}}$ are matrices of the same order, say m × n, then A + B = B + A.