Question

The matrix A = $\left[\begin{array}{cccc}a& 0& 3& 7\\ 2& 5& 1& 3\\ 0& 0& 2& 4\\ 0& 0& 0& b\end{array}\right]$ has det(A) = 100 and trace (A) = 14. The value of |a-b| is ______ .

• A. 3

Answer 1

The correct option is A 3
Given trace (A) = 14

$\Rightarrow a+7+b=14$

or a+b=7 ...(i)

Also given, det(A) =100

$\left|\begin{array}{cccc}a& 0& 3& 7\\ 2& 5& 1& 3\\ 0& 0& 2& 4\\ 0& 0& 0& b\end{array}\right|=100$

$\therefore$ 10 ab = 100
$\Rightarrow$ ab = 10 ...(ii)

Now, From (i) & (ii)
a + $\frac{10}{a}=7$
$\Rightarrow$ $a^2+10=7a$
$\Rightarrow$$a^2-7a+10=0$
$\Rightarrow$$a^2-5a-2a+10=0$
$\Rightarrow$$(a-5)-2(a-5)=0$
$\therefore$ $a=2,5$
Hence, $b=5,2$
$\therefore$ $a-b=3$