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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Matrices

NDA 2018 · Mathematics · Matrices

Consider the following in respect of matrices A and B of same...

Consider the following in respect of matrices A and B of same order: 1. A2−B2=(A+B)(A−B)2.\mathrm{A}^2-\mathrm{B}^2=(\mathrm{A}+\mathrm{B})(\mathrm{A}-\mathrm{B}) 2.(A-I)(I+A)=0 \Leftrightarrow A^2=IWhereIistheidentitymatrixandOisthenullmatrix.Whichoftheaboveis/arecorrect?Where I is the identity matrix and O is the null matrix. Which of the above is/are correct?

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

B. 2 only

Step-by-step solution

Matrix product is commutative if both are diagonal matrices of same order. ⇒A2−B2=(A+B)(A−B)\Rightarrow \quad \mathrm{A}^2-\mathrm{B}^2=(\mathrm{A}+\mathrm{B})(\mathrm{A}-\mathrm{B}) is not true. Next, (A−(\mathrm{A}-\mathrm{I})(A+)(\mathrm{A}+\mathrm{I})=0⇒A2+AI−IA−)=0\Rightarrow \quad \mathrm{A}^2+\mathrm{AI}-\mathrm{IA}-\mathrm{I}2=0(∵^2=0(\because\mathrm{AI}=\mathrm{IA})⇒A2=I)\Rightarrow \mathrm{A}^2=\mathrm{I} is correct.
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