Question

If A is a 3\times 3 matrix which has its inverse equal to A , then ${A^2}$ is
.a. .$\left[ {\begin{array}{\times {20}{c}}
  0&1&0 \\
  1&1&1 \\
  0&1&0
\end{array}} \right]$..
b. $\left[ {\begin{array}{\times {20}{c}}
  1&0&1 \\
  0&0&0 \\
  1&0&1
\end{array}} \right]$
c. $\left[ {\begin{array}{\times {20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right]$
         d. $\left[ {\begin{array}{\times {20}{c}}
  1&1&1 \\
  1&1&1 \\
  1&1&1
\end{array}} \right]$

Answer 1

Hint:
We are given that the matrix A is equal to its inverse ${A^{ - 1}} = A$ and we know that $A{A^{ - 1}} = I$.using the property given we get $A{A^{ - 1}} = {A^2}$ and equating we get the value of ${A^2}$.

Complete step by step solution:
We have a 3\times 3 matrix A
And we are given that the inverse of A is equal to A
That is
 $ \Rightarrow {A^{ - 1}} = A$ …………..(1)
We know that $A{A^{ - 1}} = I$ …………..(2)
Where I is a 3\times 3 identity matrix
Identity matrix is a matrix which has the elements in the main diagonal to be one
Now from (1) , equation (2) becomes
$ \Rightarrow A{A^{ - 1}} = AA = {A^2}$ ………..(3)
Now the left hand side of equation (2) and (3) is the same so equating their right hand side we get
$ \Rightarrow {A^2} = I$
Hence we get that ${A^2}$ is a identity matrix
$ \Rightarrow {A^2} = \left[ {\begin{array}{\times {20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right]$

The correct option is c.

Note:
1) In mathematics, a matrix (plural: matrices) is a rectangle of numbers, arranged in rows and columns. The rows are each left-to-right (horizontal) lines, and the columns go top-to-bottom
2) Every square dimension set of a matrix has a special counterpart called the "identity matrix", represented by the symbol I
3) The identity matrix has nothing but zeroes except on the main diagonal, where there are all ones.
4) An inverse matrix is a matrix that, when multiplied by another matrix, equals the identity matrix.