KCET · Maths · Matrices
If \(A\) and \(B\) are square matrices of order \(n\) such that \(A^{2}-B^{2}=(A-B)(A+B)\), then which of the following will be true?
- A Either \(A\) or \(B\) is zero matrix
- B \(A=B\)
- C \(A B=B A\)
- D Either \(A\) or \(B\) is identity matrix
Answer & Solution
Correct Answer
(C) \(A B=B A\)
Step-by-step Solution
Detailed explanation
Given,
\(\begin{aligned} A^{2}-B^{2} &=(A-B)(A+B) \\ &=A^{2}-B A+A B-B^{2} \end{aligned}\)
\(\begin{aligned} \Rightarrow & 0 &=-B A+A B \\ \Rightarrow & A B &=B A \end{aligned}\)
Also, options (a), (b) and (d) satisfy the given condition none of those is a necessary condition for \(A^{2}-B^{2}=(A-B)(A+B)\)
\(\begin{aligned} A^{2}-B^{2} &=(A-B)(A+B) \\ &=A^{2}-B A+A B-B^{2} \end{aligned}\)
\(\begin{aligned} \Rightarrow & 0 &=-B A+A B \\ \Rightarrow & A B &=B A \end{aligned}\)
Also, options (a), (b) and (d) satisfy the given condition none of those is a necessary condition for \(A^{2}-B^{2}=(A-B)(A+B)\)
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