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{x + \lambda }&x&x \\

x&{x + \lambda }&x \\

x&x&{x + \lambda }

\end{array}} \right)$ then ${A^{ - 1}}$ exists if

$(A)x \ne 0$

$(B)\lambda \ne 0$

$(C)3x + \lambda \ne 0$

$(D)x \ne 0,\lambda \ne 0$

Answer

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It is given that the question stated as a matrix as below

Let us consider, $A = $$\left( {\begin{array}{*{20}{c}}

{x + \lambda }&x&x \\

x&{x + \lambda }&x \\

x&x&{x + \lambda }

\end{array}} \right)$

We can find the inverse of the given matrix by using the formula, ${A^{ - 1}} = \dfrac{1}{{\left| A \right|}}adj{\text{ }}A$

But we can find it as, when the condition exists $\left| A \right| \ne 0$

Hence we need to show the determinant for the given matrix is non-zero.

To show: $\left| {\left( {\begin{array}{*{20}{c}}

{x + \lambda }&x&x \\

x&{x + \lambda }&x \\

x&x&{x + \lambda }

\end{array}} \right)} \right| \ne 0$ under which conditions

Firstly we have to apply the elementary transformations in the rows

${R_1} \to {R_1} - {R_2}$ and ${R_3} \to {R_3} - {R_2}$

Now to make changes in the row $1$ and row $3$ we are applied elementary operations, which means entries of the row $2$ is subtracting from the entries of the row $1$ respectively.

Similarly entries of row $2$ is subtracting from the entries of row $3$ to make changes in the row $3$respectively,

We get, $\left| {\left( {\begin{array}{*{20}{c}}

\lambda &{ - \lambda }&0 \\

x&{x + \lambda }&x \\

0&{ - \lambda }&\lambda

\end{array}} \right)} \right| \ne 0$

Expanding the matrix along ${R_1}$,

$ \Rightarrow \lambda (\lambda (x + \lambda ) + x\lambda ) - x( - {\lambda ^2} + 0) \ne 0$(For inverse)

On open brackets we multiply the terms we get,

$ \Rightarrow \lambda (\lambda x + {\lambda ^2} + x\lambda ) + x{\lambda ^2} + 0 \ne 0$

Open the brackets we multiply the terms and we get,

$ \Rightarrow x{\lambda ^2} + {\lambda ^3} + x{\lambda ^2} + x{\lambda ^2} + 0 \ne 0$

Adding the same terms we get,

$ \Rightarrow 3x{\lambda ^2} + {\lambda ^3} \ne 0$

Taking ${\lambda ^2}$ as common , we get,

$ \Rightarrow {\lambda ^2}(3x + \lambda ) \ne 0$

These two individually are not equal to zero

Separately, we put both are not equal to zero

${\lambda ^2} \ne 0$ And $3x + \lambda \ne 0$

We know when the square of the number is not equal to zero hence the number cannot be zero it is not possible.

So we can say, $\lambda \ne 0$ And $3x + \lambda \ne 0$

Make sure to take the signs right. The inverse of a matrix can be found by using row or column operations not both in a particular answer.