Question

What is a periodic matrix?

Answer 1

Hint: Here the given question’s solution will be in the form of a descriptive way. So first we explain the matrix and then the periodic matrix and we explain the periodic matrix by giving an example. Hence this will be the required solution for the given question.

Complete step-by-step answer:
Matrix refers to an ordered rectangular arrangement of numbers which are either real or complex or functions. We enclose Matrix by \[\left[ {} \right]\] or \[\left( {} \right)\].
There are different types of Matrix namely, Row Matrix, Square Matrix, Column Matrix, Rectangle Matrix, Diagonal Matrix, Scalar Matrix, Zero or Null Matrix, Unit or Identity Matrix, Upper Triangular Matrix and Lower Triangular Matrix.
Now we will consider the given question, here we have to define periodic matrix and it is defined as
Periodic Matrix: A periodic matrix is defined as a square matrix such that. for k which can be taken as any positive integer. Also, if \[k\] is the least such positive integer then the square matrix is said to be a periodic matrix with the period \[k\].
Or it can also be defined as
A square matrix \[A\] such that the matrix power \[{A^{k + 1}} = A\] for \[k\] a positive integer is called a periodic matrix. If \[k\] is the least such integer, then the matrix is said to have period \[k\]. If \[k = 1\], then \[{A^2} = A\] and \[A\] is called idempotent.
For example:
Let we discuss the example for \[{A^2} = A\], which is called idempotent.
Consider the matrix \[A = \left[ {\begin{array}{*{20}{c}}
  4&{ - 1} \\
  {12}&{ - 3}
\end{array}} \right]\]
On squaring the matrix A we have
\[ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
  4&{ - 1} \\
  {12}&{ - 3}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  4&{ - 1} \\
  {12}&{ - 3}
\end{array}} \right]\]
On multiplying
\[ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
  {16 - 12}&{ - 4 + 3} \\
  {48 - 36}&{ - 12 + 9}
\end{array}} \right]\]
On simplifying we have
\[ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
  4&{ - 1} \\
  {12}&{ - 3}
\end{array}} \right]\]
On squaring we had obtained the same matrix which we have taken.
\[ \Rightarrow {A^2} = A\]
Therefore the matrix \[A = \left[ {\begin{array}{*{20}{c}}
  4&{ - 1} \\
  {12}&{ - 3}
\end{array}} \right]\] is called a periodic matrix.

Note: This is one of the subtopic of the matrices, if the question involves another subtopic of the matrices in this way to explain the content by considering the example. If we explain the topic by considering the example then the topic will understand in a clear way.