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Let A be a square matrix of order 3 satisfies the relation `A^(3)-6A^(2)+7A-8I=O` and `B=A-2I`. Also, det. `A=8`, thenA. det. `("adj. "(I-2A^(-1))=25/16`B. adj. `((B/2)^(-1))=B/10`C. det. `("adj. "(I-2A^(-1)))=75/32`D. adj. `((B/2)^(-1))=(2B)/5` |
Answer» Given that `B=A-2I` ...(i) And `A^(3)-6A^(2)+7A-8I=O` `:. (A-2I)^(3)=5A` ...(ii) `:. B^(3)=5A` `implies |B^(3)|=|5A|=5^(3)|A|=5^(3)xx8` `implies |B|=10` Now, from (i), we get `:. A^(-1) B=I-2A^(-1)` `:. |"adj. "(I-2A^(-1))|=|I-2A^(-1)|^(2)` `=|A^(-1)B|^(2)=((|B|)/(|A|))^(2)` `=(5/4)^(2)` `=25/16` adj `((B/2)^(-1))=("adj."(B/2))^(-1)=("adj."("adj." B/2))/(|"adj." B/2|)` `=(|B/2|B/2)/(|B/2|^(2))=(B/2)/(|B/2|)=(4B)/(|B|)=(2B)/5` |
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