Answer
Verified
417.3k+ views
Hint: If a matrix is \[A = \left( {\begin{array}{*{20}{c}}
a&b \\
c&d
\end{array}} \right)\], then the inverse of the matrix is, ${A^{ - 1}} = \dfrac{1}{{\det A}}\left( {\begin{array}{*{20}{c}}
d&{ - b} \\
{ - c}&a
\end{array}} \right)$, use this to solve.
Complete step-by-step answer:
We have been given in the question, $\left( {\begin{array}{*{20}{c}}
{\dfrac{1}{{25}}}&0 \\
x&{\dfrac{1}{{25}}}
\end{array}} \right) = {\left( {\begin{array}{*{20}{c}}
5&0 \\
{ - a}&5
\end{array}} \right)^{ - 2}}$,
Now let $A = \left( {\begin{array}{*{20}{c}}
5&0 \\
{ - a}&5
\end{array}} \right)$, so we can find the inverse of the matrix by using the formula,
${A^{ - 1}} = \dfrac{1}{{\det A}}\left( {\begin{array}{*{20}{c}}
d&{ - b} \\
{ - c}&a
\end{array}} \right)$,
Therefore, $\det A = 25$,
Hence, the inverse is, ${A^{ - 1}} = \dfrac{1}{{25}}\left( {\begin{array}{*{20}{c}}
5&0 \\
a&5
\end{array}} \right)$,
So, the given equation becomes, $\left( {\begin{array}{*{20}{c}}
{\dfrac{1}{{25}}}&0 \\
x&{\dfrac{1}{{25}}}
\end{array}} \right) = {A^{ - 2}} - (1)$
Therefore, ${(A)^{ - 2}} = \dfrac{1}{{625}}{\left( {\begin{array}{*{20}{c}}
5&0 \\
a&5
\end{array}} \right)^2} = \dfrac{1}{{625}}\left( {\begin{array}{*{20}{c}}
{25}&0 \\
{10a}&{25}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{\dfrac{1}{{25}}}&0 \\
{\dfrac{{2a}}{{125}}}&{\dfrac{1}{{25}}}
\end{array}} \right)$
Now put the value of ${A^{ - 2}}$in equation (1), we get,
$
\left( {\begin{array}{*{20}{c}}
{\dfrac{1}{{25}}}&0 \\
x&{\dfrac{1}{{25}}}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{\dfrac{1}{{25}}}&0 \\
{\dfrac{{2a}}{{125}}}&{\dfrac{1}{{25}}}
\end{array}} \right) \\
\Rightarrow x = \dfrac{{2a}}{{125}} \\
$
Hence, we get the value of x as 2a/125, after solving the equations.
So, the correct answer is ${\text{B}}{\text{. }}\dfrac{{2a}}{{125}}$.
Note- Whenever such types of questions appear, be careful while finding the inverse of a matrix, use the formula, ${A^{ - 1}} = \dfrac{1}{{\det A}}\left( {\begin{array}{*{20}{c}}
d&{ - b} \\
{ - c}&a
\end{array}} \right)$, only when the matrix is a $2 \times 2$ matrix. After finding the value of ${A^{ - 2}}$, put in the given equation and solve it to find the value of x.
a&b \\
c&d
\end{array}} \right)\], then the inverse of the matrix is, ${A^{ - 1}} = \dfrac{1}{{\det A}}\left( {\begin{array}{*{20}{c}}
d&{ - b} \\
{ - c}&a
\end{array}} \right)$, use this to solve.
Complete step-by-step answer:
We have been given in the question, $\left( {\begin{array}{*{20}{c}}
{\dfrac{1}{{25}}}&0 \\
x&{\dfrac{1}{{25}}}
\end{array}} \right) = {\left( {\begin{array}{*{20}{c}}
5&0 \\
{ - a}&5
\end{array}} \right)^{ - 2}}$,
Now let $A = \left( {\begin{array}{*{20}{c}}
5&0 \\
{ - a}&5
\end{array}} \right)$, so we can find the inverse of the matrix by using the formula,
${A^{ - 1}} = \dfrac{1}{{\det A}}\left( {\begin{array}{*{20}{c}}
d&{ - b} \\
{ - c}&a
\end{array}} \right)$,
Therefore, $\det A = 25$,
Hence, the inverse is, ${A^{ - 1}} = \dfrac{1}{{25}}\left( {\begin{array}{*{20}{c}}
5&0 \\
a&5
\end{array}} \right)$,
So, the given equation becomes, $\left( {\begin{array}{*{20}{c}}
{\dfrac{1}{{25}}}&0 \\
x&{\dfrac{1}{{25}}}
\end{array}} \right) = {A^{ - 2}} - (1)$
Therefore, ${(A)^{ - 2}} = \dfrac{1}{{625}}{\left( {\begin{array}{*{20}{c}}
5&0 \\
a&5
\end{array}} \right)^2} = \dfrac{1}{{625}}\left( {\begin{array}{*{20}{c}}
{25}&0 \\
{10a}&{25}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{\dfrac{1}{{25}}}&0 \\
{\dfrac{{2a}}{{125}}}&{\dfrac{1}{{25}}}
\end{array}} \right)$
Now put the value of ${A^{ - 2}}$in equation (1), we get,
$
\left( {\begin{array}{*{20}{c}}
{\dfrac{1}{{25}}}&0 \\
x&{\dfrac{1}{{25}}}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{\dfrac{1}{{25}}}&0 \\
{\dfrac{{2a}}{{125}}}&{\dfrac{1}{{25}}}
\end{array}} \right) \\
\Rightarrow x = \dfrac{{2a}}{{125}} \\
$
Hence, we get the value of x as 2a/125, after solving the equations.
So, the correct answer is ${\text{B}}{\text{. }}\dfrac{{2a}}{{125}}$.
Note- Whenever such types of questions appear, be careful while finding the inverse of a matrix, use the formula, ${A^{ - 1}} = \dfrac{1}{{\det A}}\left( {\begin{array}{*{20}{c}}
d&{ - b} \\
{ - c}&a
\end{array}} \right)$, only when the matrix is a $2 \times 2$ matrix. After finding the value of ${A^{ - 2}}$, put in the given equation and solve it to find the value of x.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
State the differences between manure and fertilize class 8 biology CBSE
Why are xylem and phloem called complex tissues aBoth class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
What would happen if plasma membrane ruptures or breaks class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What precautions do you take while observing the nucleus class 11 biology CBSE
What would happen to the life of a cell if there was class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE