Show that a determinant is equal to this variable.

Question

Show that the :

determinant of: \begin{pmatrix}0&0&a_{13}\\0&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}

is equal to $-A_{13}A_{22}A_{31}$

I believe the cofactor and submatrix formula is to be used. This is specifically located in the chapter of finding determinants using row reduction! Please keep it simple! Thank you!

Answer 1

Going along the top row, we have

$$0(a_{22}a_{33} - a_{32}a_{23}) - 0(0 - a_{31}a_{23}) + a_{13}(0 - a_{31}a_{22}).$$

See my answer here for an explanation.

Answer 2

Hint: Expand through first row.