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!

Switch rows $1$ and $3$, then your matrix is upper triangular, with determinant $a_{13}a_{22}a_{31}$. Since you did a row switch this is the opposite of the determinant of the original matrix. — Adam Hughes, Oct 11 '14 at 20:27
i want to give adam hughes or mesel credit whoever posts that first. — Newton, Oct 11 '14 at 20:29
Where you wrote $-A_{13}A_{22}A_{31}$, did you actually mean $-a_{13}a_{22}a_{31}$? ${}\qquad{}$ — Michael Hardy, Oct 11 '14 at 21:20

score 1 · Accepted Answer · edited Apr 13 '17 at 12:21

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.

edited Apr 13 '17 at 12:21

Community

1

answered Oct 11 '14 at 20:28

score 0 · Answer 2 · answered Oct 11 '14 at 20:28

0

Hint: Expand through first row.

answered Oct 11 '14 at 20:28

Ri-Li

9,038

Show that a determinant is equal to this variable.

2 Answers2