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8. det(A) = det(At)Tutorials 5.3.6.1. Find the value ofxsuch that:(a)x-35-3x-2= 45(b)x-12x1-102-1-x= 12127
2. Use properties of determinants to evaluate:(a)3-4-26-4-1923(b)1kk21kk21kk2(c)3305220-241-3021032(d)40010333-101242322423-3-12093. Show that the value of the following determinant is independent ofx.sinxcosx0-cosxsinx0sinx-cosxsinx+ cosx1= 0.5.4Matrix InversionDefinition 5.4.1.Ann×nsquare matrixAis invertible (non-singular) if thereexist a matrixDsuch thatAD=DA=In. We sayDis an inverse matrix ofAandD=A-1.Note 5.4.2.The following facts are important:•IfA-1exists, then it is unique and square of the same order asA.128
•An invertible matrix and its inverse commute with respect to matrix mul-tiplication i.e.AA-1=A-1A=In.•A non-square matrix can never be invertible, and also among square ma-trices there are infinitely many that are not invertible.•There is a symmetric relationship between a matrix and it’s inverse; i.e.ifBis the inverse ofA, thenAis also an inverse ofB.•IfA-1exists, then the system of linear equations withAas a coefficientmatrix ALWAYS has a unique solution, whatever the right hand side.We can actually write the equation asAX=BwhereXis the columnmatrix of variablesx, y, z,etc andBis the column matrix of the righthand side. ThenAX=BimpliesA-1AX=A-1B, thenInX=A-1B,henceX=A-1B. This will yield the unique solution. (We’ll touch thisin detail later).We will study two methods of finding the inverse of a matrix. The first methodinvolvesGauss-Jordan operationand the second is theadjoint method.A.Gauss-Jordan elimination method:We will make use of the following algorithm called Matrix inversion algo-rithm in order to evaluate the inverse using the Gauss-Jordan elimination.NB:Matrix Inversion Algorithm:IfAis a (square) invertible matrix, there exists a sequence of elementaryrow operations that carryAto the identity matrixIof the same size,writtenA→I. This same series of row operations carries I toA-1; thatis,I→A-1. The algorithm can be summarized as follows:[A|I]→[I|A-1]where the row operations on A and I are carried out simultaneously.Example 5.4.3.Use the Gauss-Jordan elimination to find the inverse of129
the following matrices:(a)A=-234-5; (b)B=27114-1130(a)(A|I) =-234-51001=-23011021R2+ 2R1=-2001-5-321R1-3R2=1001523221(-12)×R1∴A-1=523221.(b)(A|I) =27114-1130100010001130
First interchange row 1 and 2.∼14-1271130010100001R2↔R1∼14-10-130-110101-201-11R2-2R1R3-R1∼14-101-30-11010-1200-11-R2∼101101-300-24-70-120-111R1-4R2R3+R2∼101101-30014-70-12012-12-12-12R3∼101101-3001-32-321121212-3212-12-12R1-11R3R2+ 3R3HenceA-1=-32-321121212-3212-12-12=12-3-31111-31-1-1B.The Adjoint Method:Definition 5.4.4.