Answer

Verified

402.9k+ views

**Hint:**To solve this question, we have to remember that the inverse of matrix A is given by, ${A^{ - 1}} = \dfrac{1}{{\left| A \right|}}adj.A$, where adj. A denotes the adjoint of matrix A and $\left| A \right|$ is the determinant of A.

**Complete step-by-step answer:**

We have,

$ \Rightarrow A = \left( {\begin{array}{*{20}{c}}

x&0&0 \\

0&y&0 \\

0&0&z

\end{array}} \right)$

We know that, for A to be invertible, $\left| A \right| \ne 0$

So, first we will find $\left| A \right|$

$ \Rightarrow x\left( {yz - 0} \right) - 0 - 0$

$ \Rightarrow \left| A \right| = xyz$

We can see that $\left| A \right| \ne 0$, hence, A is invertible.

Now, we will find the adj. A.

In order to find the adj. A, we have to find the cofactor matrix of A.

We know that,

Cofactor, \[{C_{ij}}\]of \[{a_{ij}}\] in A = \[{\left[ {{a_{ij}}} \right]_{n \times n}}\] is equal to ${\left( { - 1} \right)^{i + j}}{M_{ij}}$

Where ${M_{ij}}$ is the minor.

So,

Cofactor of \[{a_{11}}\] = $\left| {\begin{array}{*{20}{c}}

y&0 \\

0&z

\end{array}} \right| = yz$

Cofactor of \[{a_{12}}\] = $\left| {\begin{array}{*{20}{c}}

0&0 \\

0&z

\end{array}} \right| = 0$

Cofactor of \[{a_{13}}\] = $\left| {\begin{array}{*{20}{c}}

0&y \\

0&0

\end{array}} \right| = 0$

Cofactor of \[{a_{21}}\] = $\left| {\begin{array}{*{20}{c}}

0&0 \\

0&z

\end{array}} \right| = 0$

Cofactor of \[{a_{22}}\] = $\left| {\begin{array}{*{20}{c}}

x&0 \\

0&z

\end{array}} \right| = xz$

Cofactor of \[{a_{23}}\] = $\left| {\begin{array}{*{20}{c}}

x&0 \\

0&0

\end{array}} \right| = 0$

Cofactor of \[{a_{31}}\] = $\left| {\begin{array}{*{20}{c}}

0&0 \\

y&0

\end{array}} \right| = 0$

Cofactor of \[{a_{32}}\] = $\left| {\begin{array}{*{20}{c}}

x&0 \\

0&0

\end{array}} \right| = 0$

Cofactor of \[{a_{33}}\] = $\left| {\begin{array}{*{20}{c}}

x&0 \\

0&y

\end{array}} \right| = xy$

Therefore, the cofactor matrix of A is $\left( {\begin{array}{*{20}{c}}

{yz}&0&0 \\

0&{xz}&0 \\

0&0&{xy}

\end{array}} \right)$

Now, the adj. A is the transpose of the cofactor matrix of A.

Therefore,

\[adj.A = \left( {\begin{array}{*{20}{c}}

{yz}&0&0 \\

0&{xz}&0 \\

0&0&{xy}

\end{array}} \right)\]

We know that,

Inverse of A is given by, ${A^{ - 1}} = \dfrac{1}{{\left| A \right|}}adj.A$

So,

$ \Rightarrow {A^{ - 1}} = \dfrac{1}{{xyz}}\left( {\begin{array}{*{20}{c}}

{yz}&0&0 \\

0&{xz}&0 \\

0&0&{xy}

\end{array}} \right)$

Taking $\dfrac{1}{{xyz}}$ inside the matrix, we will get

\[ \Rightarrow {A^{ - 1}} = \left( {\begin{array}{*{20}{c}}

{\dfrac{{yz}}{{xyz}}}&0&0 \\

0&{\dfrac{{xz}}{{xyz}}}&0 \\

0&0&{\dfrac{{xy}}{{xyz}}}

\end{array}} \right)\]

\[ \Rightarrow {A^{ - 1}} = \left( {\begin{array}{*{20}{c}}

{\dfrac{1}{x}}&0&0 \\

0&{\dfrac{1}{y}}&0 \\

0&0&{\dfrac{1}{z}}

\end{array}} \right)\]

We can write this as:

\[ \Rightarrow {A^{ - 1}} = \left( {\begin{array}{*{20}{c}}

{{x^{ - 1}}}&0&0 \\

0&{{y^{ - 1}}}&0 \\

0&0&{{z^{ - 1}}}

\end{array}} \right)\]

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

**Note:**Whenever we asked such type of questions, we have to remember that a square matrix of order n is invertible if there exists a square matrix B of the same order such that $AB = {I_n} = BA$, in such a way, we can write ${A^{ - 1}} = B$ A square matrix is invertible if and only if it is non-singular. Through these things, we can easily solve the questions.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail