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**Hint**: In the diagonal matrix, all the elements of the matrix are zero except the diagonal running from the upper left to the lower left.

Let the elements of diagonal be \[{{a}_{ii}}\] where i = 1,2,3…. n

Substitute the values in the diagonal and solve for \[{{A}^{2}}=A\]. Simplify the result to get a relation between the result and the diagonal elements and hence calculate the number of values for matrix A.

**:**

__Complete step-by-step answer__Consider a diagonal matrix A of order n:

\[A=\left[ \begin{matrix}

{{a}_{11}} & 0 & . & . \\

0 & {{a}_{22}} & . & . \\

. & . & . & . \\

0 & . & . & {{a}_{nn}} \\

\end{matrix} \right]\]

where $ \left( {{a}_{11}},{{a}_{22}},.....,{{a}_{nn}} \right) $ are elements of diagonal.

So,

\[{{A}^{2}}=\left[ \begin{matrix}

{{a}_{11}} & 0 & . & . \\

0 & {{a}_{22}} & . & . \\

. & . & . & . \\

0 & . & . & {{a}_{nn}} \\

\end{matrix} \right]\left[ \begin{matrix}

{{a}_{11}} & 0 & . & . \\

0 & {{a}_{22}} & . & . \\

. & . & . & . \\

0 & . & . & {{a}_{nn}} \\

\end{matrix} \right]\]

\[{{A}^{2}}=\left[ \begin{matrix}

{{\left( {{a}_{11}} \right)}^{2}} & 0 & . & . \\

0 & {{\left( {{a}_{22}} \right)}^{2}} & . & . \\

. & . & . & . \\

0 & . & . & {{\left( {{a}_{nn}} \right)}^{2}} \\

\end{matrix} \right]......(1)\]

Hence, \[{{A}^{2}}={{\left( \text{Diagonal element} \right)}^{2}}\]

i.e. \[{{A}^{2}}=\left[ {{\left( {{a}_{11}} \right)}^{2}},{{\left( {{a}_{22}} \right)}^{2}},......,{{\left( {{a}_{nn}} \right)}^{2}} \right]......(2)\]

Since it is given in the question that \[{{A}^{2}}=A\]

Therefore, \[\left[ {{\left( {{a}_{11}} \right)}^{2}},{{\left( {{a}_{22}} \right)}^{2}},......,{{\left( {{a}_{nn}} \right)}^{2}} \right]=\left[ {{a}_{11}},{{a}_{22}},......{{a}_{nn}} \right]......(3)\]

We can write equation (3) as:

\[{{\left( {{a}_{11}} \right)}^{2}}={{a}_{11}};{{\left( {{a}_{22}} \right)}^{2}}={{a}_{22}};......;{{\left( {{a}_{nn}} \right)}^{2}}={{a}_{nn}}......(4)\]

Simplifying the equation (4), we get:

\[{{\left( {{a}_{ii}} \right)}^{2}}={{a}_{ii}}\], for all elements i = 1,2,3……n

Now we can write equation (4) as:

\[{{\left( {{a}_{ii}} \right)}^{2}}-{{a}_{ii}}=0\]

\[\Rightarrow {{a}_{ii}}\left( {{a}_{ii}}-1 \right)=0\]

Therefore, we get:

\[{{a}_{ii}}=0\text{ or }{{a}_{ii}}=1\], for all elements i = 1,2,3……n

For each element of diagonal, we have two choices, i.e. 0 or 1

Therefore, total values of matrix A are: \[2\times 2\times 2\times .......n\text{ }times\]

i.e. \[{{2}^{n}}\]

**So, the correct answer is “Option C”.**

**Note**: Since the diagonal matrix contains zero value of most of its elements, don’t choose option (a) directly.

Also, as given in the question: \[{{A}^{2}}=A\], there may be a possibility of getting A = 1, which neglects the other choice. So be careful with these two options given.

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