Question

Let $A=\left[\begin{array}{ccc}x+\lambda & x& x\\ x& x+\lambda & x\\ x& x& x+\lambda \end{array}\right]$, then $A^{-1}$ exists if

• A. $x\ne 0$
• B. $\lambda \ne 0$
• C. $3x+\lambda \ne 0,\lambda \ne 0$
• D. $x\ne 0,\lambda \ne 0$

Answer 1

Answer: (C)

$3x+\lambda \ne 0,\lambda \ne 0$

$A=\left[\begin{array}{ccc}x+\lambda & x& x\\ x& x+\lambda & x\\ x& x& x+\lambda \end{array}\right]$
$A^{-1}$ exists if $\left|A\right|\ne 0$
$\left|A\right|=\left|\begin{array}{ccc}x+\lambda & x& x\\ x& x+\lambda & x\\ x& x& x+\lambda \end{array}\right|\ne 0$
$R_1\to R_1+R_2+R_3$
$(3x+\lambda )\left|\begin{array}{ccc}1& 1& 1\\ x& x+\lambda & x\\ x& x& x+\lambda \end{array}\right|\ne 0$
$C_2\to C_2-C_1$ and $C_3\to C_3-C_1$
$(3x+\lambda )\left|\begin{array}{ccc}1& 0& 0\\ x& \lambda & 0\\ x& 0& \lambda \end{array}\right|\ne 0$
$(3x+\lambda )\left[{\lambda }^2\right]\ne 0$
$3x+\lambda \ne 0,\lambda \ne 0$