Question

The matrix $A=\left[\begin{array}{ccc}1& 3& 2\\ 1& x-1& 1\\ 2& 7& x-3\end{array}\right]$ will have inverse for every real number x except for

• A. $x=\frac{11\pm \sqrt{5}}{2}$
• B. $x=\frac{9\pm \sqrt{5}}{2}$
• C. $x=\frac{11\pm \sqrt{3}}{2}$
• D. $x=\frac{9\pm \sqrt{3}}{2}$

Answer 1

The correct option is A $x=\frac{11\pm \sqrt{5}}{2}$

Solution:

For $A$ have a inverse, $|A|\ne 0.$

$|A|=\left\{\right(x-1\left)\right(x-3)-7\}+3\{2-(x-3\left)\right\}+2\{7-2(x-1\left)\right\}$

$=x^2-4x+3-7+6-3x+9+14-4x+4$

$=x^2-11x+29$

$=(x-\frac{11+\sqrt{5}}{2})(x-\frac{11-\sqrt{5}}{2})$

So, $|A|\ne 0$

$x\ne \left(\frac{11\pm \sqrt{5}}{2}\right)$

i.e. for $x=\frac{11\pm \sqrt{5}}{2}$ matrix $A$ will not have its inverse.

Hence, A is the correct option.