# Thread: Determinant of 4 by 4 matrix

1. ## Determinant of 4 by 4 matrix

Hi,
I have a problem with understanding how to calculate a determinant of 4 by 4 matrix with no zero elements.
If given matrix is:
$
A= \begin{pmatrix}a_{11} & a_{12} & a_{13} & a_{14}\\
a_{21} & a_{22} & a_{23} & a_{24}\\a_{31} & a_{32} & a_{33} & a_{34}\\a_{41} & a_{42} & a_{43} & a_{44}\end{pmatrix}$

Find determinant.

My try is:
$
det A= a_{11} a_{22} a_{33} a_{44}- a_{11} a_{22} a_{34} a_{43}
-a_{11} a_{23} a_{32} a_{44} +a_{11} a_{23} a_{34} a_{42}$

$
+a_{11} a_{24} a_{32} a_{43} -a_{11} a_{24} a_{33} a_{42}
-a_{12} a_{21} a_{33} a_{44} +a_{12} a_{21} a_{34} a_{43}
$

$
+a_{12} a_{23} a_{31} a_{44} +a_{12} a_{23} a_{34} a_{41}
+a_{12} a_{24} a_{31} a_{43} -a_{12} a_{24} a_{33} a_{41}
$

$
+a_{13} a_{21} a_{32} a_{44} -a_{13} a_{21} a_{34} a_{42}
-a_{13} a_{22} a_{31} a_{44} +a_{13} a_{22} a_{34} a_{41}$

$
+a_{13} a_{24} a_{31} a_{42} -a_{13} a_{24} a_{32} a_{41}
-a_{14} a_{21} a_{32} a_{43} +a_{14} a_{21} a_{33} a_{42}$

$
-a_{14} a_{22} a_{31} a_{43} -a_{14} a_{22} a_{33} a_{41}
+a_{14} a_{23} a_{31} a_{42} -a_{14} a_{23} a_{32} a_{41}
$

Is this a correct answer?

2. Hello, Bernice!

I did not check your answer . . . too much work!

$A\;= \;\begin{pmatrix}a_{11} & a_{12} & a_{13} & a_{14}\\
a_{21} & a_{22} & a_{23} & a_{24}\\a_{31} & a_{32} & a_{33} & a_{34}\\a_{41} & a_{42} & a_{43} & a_{44}\end{pmatrix}$
. . Find determinant.
Can I assume you know the "Cofactor" method?

$\text{det }\!\!A \;=\;a_{11}\begin{vmatrix}a_{22}&a_{23}&a_{24} \\ a_{32}&a_{33}&a_{34} \\ a_{42}&a_{43}&a_{44} \end{vmatrix} - a_{12}\begin{vmatrix}a_{21}&a_{23}&a_{24}\\a_{31}& a_{33}&a_{34}\\a_{41}&a_{43}&a_{44} \end{vmatrix}$ . $+\:a_{13}\begin{vmatrix}a_{21}&a_{22}&a_{24}\\a_{3 1}&a_{32}&a_{34} \\ a_{41}&a_{42}&a_{44}\end{vmatrix} - a_{14}\begin{vmatrix}a_{21}& a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\a_{41}&a_{42} &a_{43} \end{vmatrix}$

. . . $= \;a_{11}\bigg( a_{22}\begin{vmatrix}a_{33}&a_{34}\\a_{43}&a_{44}\ end{vmatrix} - a_{23}\begin{vmatrix}a_{32}&a_{34}\\a_{42}&a_{44}\ end{vmatrix} + a_{24}\begin{vmatrix}a_{32}&a_{33}\\a_{42}&a_{43}\ end{vmatrix}
\bigg)$

. . . . . $-a_{12}\bigg(a_{21}\begin{vmatrix}a_{33}&a_{34}\\a_ {43}&a_{44}\end{vmatrix} -a_{23}\begin{vmatrix}a_{31}&a_{34}\\a_{41}&a_{44}\ end{vmatrix} + a_{24}\begin{vmatrix}a_{31}&a_{33}\\a_{41}&a_{43} \end{vmatrix} \bigg)$

. . . . . . . $+a_{13}\bigg(a_{21}\begin{vmatrix}a_{32}&a_{34}\\a _{42}&a_{44}\end{vmatrix} -a_{22}\begin{vmatrix}a_{31}&a_{34}\\a_{41}&a_{44}\ end{vmatrix} + a_{24}\begin{vmatrix}a_{31}&a_{32}\\a_{41}&a_{42}\ end{vmatrix}\bigg)$

. . . . . . . . . $-a_{14}\bigg(a_{21}\begin{vmatrix}a_{32}&a_{33}\\a_ {42}&a_{43} \end{vmatrix} - a_{22}\begin{vmatrix}a_{31}&a_{33}\\a_{41}&a_{43}\ end{vmatrix} + a_{23}\begin{vmatrix}a_{31}&a_{32}\\a_{41}&a_{42}\ end{vmatrix} \bigg)$

. . . and so on.