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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES1 Mark
Question
Choose the correct answer from the given four options.
If A is a square matrix such that A2 = I, then (A - I)3 + (A + I)3 - 7A is equal to:
  1. A
  2. I - A
  3. I + A
  4. 3A
✓

Answer

  1. A

Solution:

We have, A2 = I

Now, (A - I)3 + (A + I)3 - 7A = [(A - I) + (A + I)][(A - I)2 + (A + I)2 - (A - I)(A + I)] - 7A

$[\because$ a3 + b3 = (a + b)(a2 + b2 - ab)$]$

= [(2A){A2 + I2 - 2AI + A2 + I2 + 2AI - (A2 - I2)}] - 7A

= [(2A){AI + I2 - 2AI + AI + I2 + 2AI - AI +I2}] - 7A $[\because$ A2 = AI$]$

= 2A[I + I2 + I + I2 - I + I2] - 7A

= 2A[5I - I] - 7A

= 8AI - 7AI $[\because$ A = AI$]$

= AI

= A

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