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Matrices and Determinants question

2006 · Shift 0 · Q59
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Matrices and Determinants question

2006 · Shift 0 · Q59

JEE MainMathematicsMatrices and DeterminantsMCQ+4 / −1
If AAA and BBB are square matrices of size n × nn\, \times \,nn×n such that A2−B2=(A−B)(A+B),{A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),A2−B2=(A−B)(A+B), then which of the following will be always true?
  1. A
    A=BA=BA=B
  2. B
    AB=BAAB=BAAB=BA
  3. C
    either of AAA or BBB is a zero matrix
  4. D
    either of AAA or BBB is identity matrix
View written solutionFree

Correct answer: B

  1. We are given A2−B2=(A−B)(A+B).A^2-B^2=(A-B)(A+B).A2−B2=(A−B)(A+B).

  2. Expand the right-hand side using matrix multiplication: (A−B)(A+B)=A2+AB−BA−B2.(A-B)(A+B)=A^2+AB-BA-B^2.(A−B)(A+B)=A2+AB−BA−B2.

  3. Compare this with the left-hand side: A2−B2.A^2-B^2.A2−B2. So we must have A2−B2=A2+AB−BA−B2.A^2-B^2=A^2+AB-BA-B^2.A2−B2=A2+AB−BA−B2.

  4. Cancel A2A^2A2 and −B2-B^2−B2 from both sides: 0=AB−BA.0=AB-BA.0=AB−BA. Hence, AB=BA.AB=BA.AB=BA.

  5. Therefore, the given condition implies that AAA and BBB commute.

  6. Now check the options:

  • A: A=BA=BA=B — not necessary.
  • B: AB=BAAB=BAAB=BA — always true.
  • C: either AAA or BBB is zero matrix — not necessary.
  • D: either AAA or BBB is identity matrix — not necessary.

Hence the correct option is B.\boxed{B}.B​.

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