Question

Let $f(a,b)=\left|\begin{array}{ccc}a& a^2& 0\\ 1& 2a+b& (a+b{)}^2\\ 0& 1& a+2b\end{array}\right|$,then

• A. $(a+b)$ is a factor of $f(a,b)$.
• B. $a+2b$ is a factor of $f(a,b)$
• C. $b+2a$ is a factor of $f(a,b)$
• D. $ab$ is a factor of $f(a,b)$

Answer 1

Answer: (A), (D)

$(a+b)$ is a factor of $f(a,b)$.
$ab$ is a factor of $f(a,b)$

f(a, b) $=\left|\begin{array}{ccc}a& a^2& 0\\ 1& 2a+b& (a+b{)}^2\\ 0& 1& a+2b\end{array}\right|$
$C_2\to C_2-a{C}_1$
$f(a,b)=\left|\begin{array}{ccc}a& 0& 0\\ 1& a+b& (a+b{)}^2\\ 0& 1& a+2b\end{array}\right|$
$=a\left(\right(a+b\left)\right(a+2b)-(a+b{)}^2)$
$=a(a^2+3ab+2b^2-a^2-2ab-b^2)$
$=a(ab+b^2)$
$=ab(a+b)$
Hence, both $ab$ and $a+b$ are factors of the given determinant.