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Question
If A = `[[1 0],[0 1]]`,B`[[1 0],[0 -1]]`
and C= `[[0 1],[1 0]]`
, then show that A2 = B2 = C2 = I2.
Solution
Here,
`A^(2)=A A`
`⇒ A^(2)=[[1 0 ],[0 1]]` ` [[1 0],[1 0]]`
`⇒A^(2)=[[1+0 0+1],[0+0 0+1]]`
`⇒A^(2)=[[1 0],[0 1]]`.............(1)
`B^(2)=BB`
`⇒B^(2)=[[1 0 ],[0 -1]]``[[1 0],[0 -1]]`
`⇒B^(2)=[[1+0 0-0],[0-0 0+1]]`
`⇒B^(2)=[[1 0],[0 1]]`.............(2)
`C^(2)= C C`
`⇒B^2=[[0 1 ],[1 0]]``[[0 1],[1 0]]`
`⇒B^2=[[0+1 0+0],[0+0 1+0]]`
`⇒B^2=[[1 0],[0 1]]`.............(3)
We know,
`I_2=[[1 0],[0 1]]`...........................(4)
⇒`A^2=B^2=C^2=I_2` [From eqs. (1), (2), (3) and (4)]
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