DETERMINANTS · Part 2

shown below. Consider the determinant of square matrix A = [ a ij ] × i.e., | A | = Expansion along first Row (R ) Step Multiply first element a of R by (– ) ( + ) [(– ) sum of suffixes in a ] and with the second order determinant obtained by deleting the elements of first row (R ) and first column (C ) of | A | as a lies in R and C , i.e., (– ) + a Step Multiply 2nd element a of R by (– ) + [(– ) sum of suffixes in a ] and the second order determinant obtained by deleting elements of first row (R ) and 2nd column (C ) of | A | as a lies in R and C , i.e., (– ) + a Step Multiply third element a of R by (– ) + [(– ) sum of suffixes in a ] and the second order determinant obtained by deleting elements of first row (R ) and third column (C ) of | A | as a lies in R and C , i.e., (– ) + a Step Now the expansion of determinant of A, that is, | A | written as sum of all three terms obtained in steps , and above is given by det A = |A| = (– ) + a (– ) (– ) |A| = a ( a a – a a ) – a ( a a – a a ) + a ( a a – a a ) DETERMINANTS = a a a – a a a – a a a + a a a + a