Question

Show that $\left|\begin{array}{ccc}a^{2}+2a&2a+1&1\\2a+1&a+2&1\\3&3&1\end{array}\right|$ = (a - 1)³

Answer 1

Hi.

It is a question of Determinant Evaluation.

I have used two techniques in this question.

• The Properties of the Determinant
• The Expansion Method

The benefit of using "The Properties of Determinant" is that the expansion of determinant becomes easier and shorter.

I have solved it in detail.

Kindly see the Attachment for the detailed answer.

Hope, it will help You.

Answer 2

$\text{Find the determinant of } \left|\begin{array}{ccc}a^{2}+2a&2a+1&1\\2a+1&a+2&1\\3&3&1\end{array}\right| :$

$\text{Determinant = } (a^2 + 2a)[ (a + 2) - (1)(3) ] - (2a + 1)[ (2a + 1) - (1)(3) ] + 1[3(2a + 1)- 3(a + 2)]$

$\text{Determinant = } (a^2 + 2a)[a -1 ] - (2a + 1)[ 2a -2 ] + 1[3a - 3]$

$\text{Determinant = }a^3 - a^2 + 2a^2 - 2a - (4a^2 - 4a + 2a - 2) + (3a - 3)$

$\text{Determinant = }a^3 - a^2 + 2a^2 - 2a - 4a^2 + 4a - 2a + 2 + 3a - 3$

$\text{Determinant = }a^3 - 3a^2 + 3a - 1$

$\text{Determinant = }(a - 1)(2a^2 - 2a + 1)$

$\text{Determinant = }(a - 1)(a -1)^2$

$\text{Determinant = }(a - 1)^3$ {shown)