Question

Find the value of the
determinants...
​

Answer 1

Solution:-

$\implies\begin{vmatrix}x+\lambda & x & x\\x & x+\lambda & x \\x& x & x+\lambda \end{vmatrix}$

Now we can use this

$\rm \: C_1 \to \: C_1 + C_2 + C_3$

We will get

$\implies\begin{vmatrix}3x+\lambda & x & x\\3x + \lambda & x+\lambda & x \\3x + \lambda& x & x+\lambda \end{vmatrix}$

Take common

$3x + \lambda$

we get

$\implies3x + \lambda\begin{vmatrix}1 & x & x\\1& x+\lambda & x \\1& x & x+\lambda \end{vmatrix}$

Now

$\rm \: R_2 \to \: R_2 - R_1 \\ \rm \: R_3 \to \: R_ 3 - R_1$

we get

$\implies3x + \lambda\begin{vmatrix}1 & x & x\\0& \lambda & 0\\0& 0 & \lambda \end{vmatrix}$

Taking determinant

$\rm \: C_1 \: \: \: and \: \: \: R_1$

we get

$= 3x + \lambda \times 1 ( { \lambda}^{2} - 0)$

so answer is

$\Delta = { \lambda}^{2} (3x + \lambda)$