Determinant of a 3 x 3 Matrix

The value of the determinant of a square matrix of order n ( ≥ 2) is defined as the sum of

the products of the elements of any row (respectively column) with their corresponding

co-factors.

Let

$A=\begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix}$ be a square matrix of order 3.
Using the first row, we have

$|A|=a_{11}A_{11}+a_{12}A_{12}+a_{13}A_{13}$

$|A|=a_{11}(-1)^{1+1}\begin{vmatrix} a_{22} & a_{23}\\ a_{32} & a_{33} \end{vmatrix}+a_{12}(-1)^{1+2}\begin{vmatrix} a_{21} & a_{23}\\ a_{31} & a_{33} \end{vmatrix}+a_{13}(-1)^{1+3}\begin{vmatrix} a_{21} & a_{22}\\ a_{31} & a_{32} \end{vmatrix}$

$=a_{11}(a_{22}a_{33}-a_{32}a_{23})-a_{12}(a_{21}a_{33}-a_{31}a_{23})+a_{13}(a_{21}a_{32}-a_{31}a_{22})$
We say that |A| has been expanded along $R_{1}$ . If |A| is expanded along any other row
(or column), same result would be obtained.

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Determinant of a 3 x 3 Matrix

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