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3 and 4 . determinant and metrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths3 and 4 . determinant and metrices1 Mark
MCQ
Let $p$ a non singular matrix $1 + p + {p^2} + .... + {p^n} = O$ ($O$ denotes the null matrix), then ${p^{ - 1}} = $
  • ✓
    ${p^n}$
  • B
    $ - {p^n}$
  • C
    -$(1 + p + .... + {p^n})$
  • D
    None of these
✓

Answer

Correct option: A.
${p^n}$
a
(a) $1 + p + {p^2} + .......... + {p^n} = O$
Pre-multiplying both sides by ${p^{ - 1}}$
==> ${p^{ - 1}} + I + Ip + ............ + {p^{n - 1}}I = O.{p^{ - 1}}$
==> ${p^{ - 1}} + I(1 + p + {p^2} + ......... + {p^{n - 1}}) = O$
==> ${p^{ - 1}} = - (1 + p + {p^2} + ......... + {p^{n - 1}})I$
==> ${p^{ - 1}} = {p^n}$

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