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(ii) Add the products along the three full diagonals that extend from upperleft to lower right.(iii) Subtract the products along the three full diagonals that extend fromlower left to upper right.Let’s work the previous example, now using “Sarrus scheme”:E.g.Find-23-14-7112-5Solution:detA=-23-14-7112-5=-23-14-7112-5-234-712= (-2)(-7)(-5) + (3)(1)(1) + (-1)(4)(2)-(1)(-7)(-1)-(2)(1)(-2)-(-5)(4)(3)=-70 + 3-8-7 + 4 + 60 =-18.NB:Sarrus rule does not work forn×nmatrices wheren≥4.Properties of determinants:1. The sign of a determinant changes if any two columns or rows are inter-changed.123
E.g.From an example, we saw that-23-14-7112-5=-18.If we interchange row 1 and row 3, the new determinant;12-54-71-23-1= 18.Interchanging of columns has the same effect.2. The determinant of a matrix with a zero column or zero row is zero.E.g.10-5401-20-1= 0 and1200= 0.3. If any two rows or columns of a matrix are equal, its determinant is zero.E.g.12-58-5-28-5-2= 0 sinceR2=R3.1241-3-55-3-40-5-421-52= 0 sinceC1=C4.4. If any row or column of a matrix is multiplied by a non-zero constantk,it’s determinant is multiplied byk.E.g.IfA=-23-14-7112-5,then if we multiply column 3 by -5, i.e. we have new matrixB=-2354-7-51225,124
then detB=-5 detA=-5(-18) = 90.Suppose we multiply the whole matrixAby 2, then each row will bemultiplied by two. Sodet 2A= 2×2×2 detA= 8×(-18) =-144.This is because each row or column is a multiplied by 2, hence the multipleof each row multiplies the detA, in this case three 2’s since we have 3 rowsor columns.In general IfAis ann×nmatrix, then det(kA) =kndetA.5. If any row or column of a matrix is multiplied by a non-zero constantkand the result is added to another row or column, it’s determinant remainsthe same. We can show this by performing row operation.E.g.A=-23-14-7112-5using row operation.detA=-23-14-7112-5=07-110-15211-5-2R1+ 2R3R2-4R3Expanding alongC1= 1×7-11-1521= 7×21-(-15)(-11) = 147-165 =-18.Which still yields the same result.NB:Care must be taken when performing row operation as to which rowwe multiply.Performing row operation to find the determinant is quite helpful when125
we deal with higher order square matrices, e.g. 4×4, 5×5, etc.E.g.Consider a 4×4 matrixD=13573135531375311357313553137531=13570-8-12-160-12-24-320-16-32-48R2-3R1R3-5R1R4-7R1= 1×-8-12-16-12-24-32-16-32-48= (-4)(-4)(-16)234368123=-2560-1-200-1123R1-2R3R2-3R3=-256×1×-1-20-1=-256×(1-0) =-256.126