This matrix is now in a row echelon form.(b) Now to obtain a reduced row echelon form, we continue from the last step in(a) to ensure that all columns with leading one have zeros elsewhere as follows:11101001100-1000103000000R1-R3R2-R3∼10001001100-1000103000000R1-R2Tutorials 5.2.6.Reduce the following matrix and leave your answer in(a) Row echelon form.(b) Reduced row echelon form.1.12-3-24125-8-164147528.2.12121224355736491011124369.5.3DeterminantGiven any square matrixA, there is a special scalar called the determinant ofAdenoted by det(A) or we make use of vertical bars|A|. This number is alsoimportant in determining whether a matrix isinvertibleor not, which will bestudied in the next section.We will show how to evaluate determinant of square matrices of different orders.120
(a):1×1matrix:IfA= (a), then we simply say detA=a.Example 5.3.1.A= (2) gives detA= 2.(b):2×2matrix:IfA=abcd;The determinant is defined by: detA=abcd=ad-bcExample 5.3.2.IfA=5-16-3,thendetA=5-16-3= (5)(-3)-6(-1) =-15 + 6 =-9.Definition 5.3.3.The(i, j)-th minor of a matrixA, denoted bymij(A), is thedeterminant obtained by deleting rowiand columnjofA.e.g. IfA=a11a12a13a21a22a23a31a32a33, thenm32(A) =a11a13a21a23.Definition 5.3.4.The(i, j)-th cofactor of a matrixA, denoted bycij(A), isthe signed minor given bycij(A) = (-1)i+jmij(A)ofA.e.g. IfA=a11a12a13a21a22a23a31a32a33, thenc23(A) = (-1)2+3a11a12a31a32=-a11a12a31a32.121
(c):3×3matrix:(i)Laplace expansion:GivenA=a11a12a13a21a22a23a31a32a33;detA=a11a12a13a21a22a23a31a32a33=a11(-1)1+1m11+a12(-1)1+2m12+a13(-1)1+3m13=a11(-1)2a22a23a32a33+a12(-1)3a21a23a31a33+a13(-1)4a21a22a31a32.This is Laplace expansion along row one.This expansion can be donealong any other row or column without affecting the value of the determi-nant.Example 5.3.5.A=-23-14-7112-5detA=-23-14-7112-5Expand along row 1= (-2)(-1)1+1-712-5+ 3(-1)1+2411-5+ (-1)(-1)1+34-712= (-2)(35-2)-3(-20-1)-1(8 + 7) =-66 + 63-15 =-18.(ii)Sarrus scheme:122
Another useful method which can be used to evaluate the determinant of a3×3 matrix is what we call “Sarrus scheme” or also commonly referred to as“butterfly method”. To find the det(A) using this method we follow the followingsteps:(i) Rewrite the 1stand 2ndcolumns on the right (as Columns 4 and 5).