Show that a determinant equals the product of another determinant and a polynomial function without calculating

Question

Show without calculating the determinant, that

$$ \det\left(\begin{bmatrix} a_{1}+b_{1}x & a_{1}-b_{1}x & c_{1}\\ a_{2}+b_{2}x & a_{2}-b_{2}x & c_{2}\\ a_{3}+b_{3}x & a_{3}-b_{3}x& c_{3}\\ \end{bmatrix}\right) = -2x \det\left(\begin{bmatrix} a_{1} & b_{1}x & c_{1}\\ a_{2} & b_{2}x & c_{2}\\ a_{3} & b_{3}x& c_{3}\\ \end{bmatrix}\right) $$

I've tried using the multilinearity of the determinant, but i didn't get very far and now im stuck.

Answer 1

$ \left|\begin{matrix} a_1+b_1x & a_1-b_1x & c_1\\ a_2+b_2x & a_2-b_2x & c_2\\ a_3+b_3x & a_3-b_3x & c_3 \end{matrix}\right|$

$=\;\; \left|\begin{matrix} a_1 & a_1-b_1x & c_1\\ a_2 & a_2-b_2x & c_2\\ a_3 & a_3-b_3x & c_3 \end{matrix}\right|+\left|\begin{matrix} b_1x & a_1-b_1x & c_1\\ b_2x & a_2-b_2x & c_2\\ b_3x & a_3-b_3x & c_3 \end{matrix}\right|$

$=\;\; \left|\begin{matrix} a_1 & a_1 & c_1\\ a_2 & a_2 & c_2\\ a_3 & a_3 & c_3 \end{matrix}\right|+\left|\begin{matrix} a_1 & -b_1x & c_1\\ a_2 & -b_2x & c_2\\ a_3 & -b_3x & c_3 \end{matrix}\right|+\left|\begin{matrix} b_1x & a_1 & c_1\\ b_2x & a_2 & c_2\\ b_3x & a_3 & c_3 \end{matrix}\right|+\left|\begin{matrix} b_1x & -b_1x & c_1\\ b_2x & -b_2x & c_2\\ b_3x & -b_3x & c_3 \end{matrix}\right| $

$=\;\; 0+\left|\begin{matrix} a_1 & -b_1x & c_1\\ a_2 & -b_2x & c_2\\ a_3 & -b_3x & c_3 \end{matrix}\right|+\left|\begin{matrix} b_1x & a_1 & c_1\\ b_2x & a_2 & c_2\\ b_3x & a_3 & c_3 \end{matrix}\right|+0 $

$=\;\; -x \left|\begin{matrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{matrix}\right|+ x\left|\begin{matrix} b_1 & a_1 & c_1\\ b_2 & a_2 & c_2\\ b_3 & a_3 & c_3 \end{matrix}\right| $

$=\;\; -x \left|\begin{matrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{matrix}\right|-x \left|\begin{matrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{matrix}\right| $

$=\;\; -2x \left|\begin{matrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{matrix}\right| $

Answer 2

It is indeed multilinearity you want to use. Write $$ a = \begin{bmatrix} a_{1}\\ a_{2}\\ a_{3} \end{bmatrix}, \quad b = \begin{bmatrix} b_{1}\\ b_{2}\\ b_{3} \end{bmatrix}, \quad c = \begin{bmatrix} c_{1}\\ c_{2}\\ c_{3} \end{bmatrix}, $$ then \begin{align*} \det(\begin{bmatrix}a + b x & a - b x & c\end{bmatrix}) &= \det(\begin{bmatrix}a & a & c\end{bmatrix}) - x^{2} \det(\begin{bmatrix}b & b & c\end{bmatrix}) \\&- x \det(\begin{bmatrix}a & b & c\end{bmatrix}) + x \det(\begin{bmatrix}b & a & c\end{bmatrix}). \end{align*}