What are the factors of the matrix M?

Question

A matrix is given by

$M = \begin{bmatrix} 1 & a & bc\\ 1 & b & ca\\ 1 & c & ab \end{bmatrix}$

What are the factors of this matrix M ?

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My Try :-

Multiply $R1$ by $a$, $R2$ by $b$ and $R3$ by $c$. After that, take common $abc$.

$abc\begin{bmatrix} a & a^2 & 1\\ b & b^2 & 1\\ c & c^2 & 1 \end{bmatrix}$

So, I think $abc$ will be one of the factors of this matrix $M$

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Instead of this method, I tried by one more method applying row operations .

$R2\Rightarrow R2-R1$ and $R3\Rightarrow R3-R1$. This gives

\begin{bmatrix} 1 & a & bc\\ 0 & b-a & c(a-b)\\ 0 & c-b & a(b-c) \end{bmatrix}

Hence, Determinant is $\left ( a-b \right )\left ( b-c \right )\left ( c-a \right )$, which says $(a-b)$ or $(b-c)$ or $(c-a)$ is the factor.

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Hence, All the factors completely different by the two different methods. Where am I going wrong ?

Answer 1

Hint:

$15i+30j+45k=5(3i+6j+9k)$

$15i+30j+45k=3(5i+10j+15k)$

What I meant to say is: if factors are different,then elements of matrix are also different,when multiplied,gives same matrix.

Answer 2

I cannot believe that it took MSE almost seven years to find OP's mistakes in the row operations.

The determintant of the matrix \begin{bmatrix} 1 & a & bc\\ 1 & b & ca\\ 1 & c & ab \end{bmatrix} is without doubt $(a-b)(c-a)(b-c)\,.$

With row operations,

$$\det \begin{bmatrix} 1 & a & bc\\ 0 & b-a & c(a-b)\\ 0 & c-\color{red}a & \color{red}{b(a-c)} \end{bmatrix}=b(b-a)(a-c)-c(c-a)(a-b)=(a-b)(c-a)(b-c) $$ as it must.