Question

The matrix $A = \left[ {\begin{array}{*{20}{c}}
  { - 5}&{ - 8}&0 \\
  3&5&0 \\
  1&2&{ - 1}
\end{array}} \right]$ is
A) Idempotent Matrix
B) Involutory Matrix
C) Nilpotent Matrix
D) None of these

Answer 1

Hint: Idempotent Matrix ${A^2} = A$ , Involutory Matrix ${A^2} = I$, Nilpotent Matrix ${A^k} = 0,{\text{for some }}k$.

Complete step-by-step answer:

Consider, $A = \left[ {\begin{array}{*{20}{c}}

  { - 5}&{ - 8}&0 \\

  3&5&0 \\

  1&2&{ - 1}

\end{array}} \right]$. We’ll compute ${A^2},{A^3}....$ and check which one is the correct option.

$\

  {A^2} = \left[ {\begin{array}{*{20}{c}}

  { - 5}&{ - 8}&0 \\

  3&5&0 \\

  1&2&{ - 1}

\end{array}} \right]\left[ {\begin{array}{*{20}{c}}

  { - 5}&{ - 8}&0 \\

  3&5&0 \\

  1&2&{ - 1}

\end{array}} \right] \\

   \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}

  {25 - 24 + 0}&{40 - 40 + 0}&{0 + 0 + 0} \\

  { - 15 + 15 + 0}&{ - 24 + 25 + 0}&{0 + 0 + 0} \\

  { - 5 + 6 - 1}&{ - 8 + 10 - 2}&{0 + 0 + 1}

\end{array}} \right] \\

   \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}

  1&0&0 \\

  0&1&0 \\

  0&0&1

\end{array}} \right] \\

   \Rightarrow {A^2} = I \\

\ $

Since, ${A^2} = I$ so the given matrix A is Involutory Matrix.

Option B is the correct option.

Note: In mathematics, an involutory matrix is a matrix that is its own inverse. Involutory matrices are all square roots of the identity matrix.