For two 3 × 3 matrices A and B , let A + B = 2 B ' and 3 A + 2 B = I 3 , where B ′ is the…

For two 3×3 matrices A and B, let A+B=2B' and 3A+2B=I3, where B′ is the transpose of B and I3 is 3×3 identity matrix. Then :
  1. 10A+5B=3I3
  2. 3A+6B=2I3
  3. 5A+10B=2I3
  4. B+2A=I3

Solution

Given A+B=2 B T

⇒ AT+BT=2B

⇒     B=AT+BT2

=A+BT+AT2=2BT

2A+AT=3BT

⇒     A=3BT-AT2

3A+2B=I3 ......i

⇒      33BT-AT2+2AT+BT2=I3

⇒    3BT+2BT2+2AT+BT2=I3

⇒      11BT-AT=2I3 .......ii

Equation i+ii

35B=7I3

⇒ B=I35

11 I35-A=2I3

⇒     11I35-2I3=A

⇒    A=I35

∵      5A=5B=I3

⇒       10A+5B=3I3

Asked in: JEE Main 2017 (09 Apr Online)

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