Question

Matrices help. *question attached below* Will give medal

Answer 1

So this question came on one of my exams and I was unable to do it because I did not understand a thing. Can someone please help me understand by walking me through it?

Answer 2

Have you performed the operation they suggest?

Answer 3

No, I was unable to do that because I did not know if I should end up with a 3x3 or a 2x2 matrix?

Answer 4

I meant the column subtraction.
\[\begin{align*}|M|&=
\begin{vmatrix}1&\color{red}1&\color{blue}1\\
bc&\color{red}{ca}&\color{blue}{ab}
\\b+c&\color{red}{c+a}&\color{blue}{a+b}\end{vmatrix}\\\\
&=\begin{vmatrix}1-\color{red}1&\color{red}1-\color{blue}1&\color{blue}1\\
bc-\color{red}{ca}&\color{red}{ca}-\color{blue}{ab}&\color{blue}{ab}\\
(b+c)-\color{red}{(c+a)}&\color{red}{(c+a)}-\color{blue}{(a+b)}&\color{blue}{a+b}\end{vmatrix}\\\\
&=\begin{vmatrix}0&0&1\\
c(b-a)&a(c-b)&ab\\
b+a&c+b&a+b\end{vmatrix}
\end{align*}\]

Answer 5

Ohhh :O

Answer 6

Was it really that simple?

Answer 7

What steps should I take to factorise |M| ?

Answer 8

Not sure what is intended by "factorise" here. Maybe it means take the cofactor expansion? Does that sound familiar? I think that's what you're supposed to do here due to all those 0s in the first row. A cofactor expansion along the first row reduces the above determinant to the determinant of a 2x2 submatrix, with
\[|M|=\begin{vmatrix}c(b-a)&a(c-b)\\b+a&c+b\end{vmatrix}\]

Answer 9

Yes, cofactor does sound familiar