Determinants

Determinant of a matrix of order three can be determined by expressing it in terms of second order determinants. This is known as expansion of a determinant along a row (or a column). There are six ways of expanding a determinant of order 3 corresponding to each of three rows \((R_{1}\), \(R_{2}\) and \(R_{3}\)) and three columns \((C_{1}\), \(C_{2}\) and \(C_{3})\) giving the same value as shown below.

Consider the determinant of square matrix \(A=[a_{ij}]_{3\times3}\) i.e., \(|A|=|\begin{smallmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{smallmatrix}|\)

Expansion along first Row (R1)

Step 1 Multiply first element \(a_{11}\) of \(R_{1}\) by \((-1)^{(1+1)}[(-1)^{\text{sum of suffixes in } a_{11}}]\) and with the second order determinant obtained by deleting the first row and first column \((C_{1})\) of \(|A|\) as \(a_{11}\) lies in \(R_{1}\) and \(C_{1}\), i.e., \((-1)^{1+1}a_{11}|\begin{smallmatrix}a_{22}&a_{23}\\ a_{32}&a_{33}\end{smallmatrix}|\)

Step 2 Multiply 2nd element \(a_{12}\) of \(R_{1}\) by \((-1)^{1+2}[(-1)^{\text{sum of suffixes in } a_{12}}]\) and the second order determinant obtained by deleting the first row \((R_{1})\) and second column \((C_{2})\) of \(|A|\) as \(a_{12}\) lies in \(R_{1}\) and \(C_{2},\) i.e., \((-1)^{1+2}a_{12}|\begin{smallmatrix}a_{21}&a_{23}\\ a_{31}&a_{33}\end{smallmatrix}|\)

Step 3 Multiply 3rd element of \(R_{1}\) by \((-1)^{1+3}[(-1)^{\text{sum of suffixes in } a_{13}}]\) and the second order determinant obtained by deleting the first row \((R_{1})\) and third column \((C_{3})\) of \(|A|\) as \(a_{13}\) lies in \(R_{1}\) and \(C_{3}\). i.e., \((-1)^{1+3}a_{13}|\begin{smallmatrix}a_{21}&a_{22}\\ a_{31}&a_{32}\end{smallmatrix}|\)

Step 4 Finally, the expansion of the determinant of A, that is, \(|A|\) written as the sum of all three terms obtained in steps 1, 2 and 3 above is given by \(|A|=(-1)^{1+1}a_{11}|\begin{smallmatrix}a_{22}&a_{23}\\ a_{32}&a_{33}\end{smallmatrix}|+(-1)^{1+2}a_{12}|\begin{smallmatrix}a_{21}&a_{23}\\ a_{31}&a_{33}\end{smallmatrix}| +(-1)^{1+3}a_{13}|\begin{smallmatrix}a_{21}&a_{22}\\ a_{31}&a_{32}\end{smallmatrix}|\) or \(|A|=a_{11}(a_{22}a_{33}-a_{32}a_{23})-a_{12}(a_{21}a_{33}-a_{31}a_{23}) +a_{13}(a_{21}a_{32}-a_{31}a_{22})\) \(=a_{11}a_{22}a_{33}-a_{11}a_{32}a_{23}-a_{12}a_{21}a_{33}+a_{12}a_{31}a_{23}+a_{13}a_{21}a_{32}\) \(-a_{13} a_{31} a_{22}\)