5.3 Determinants and Cramer’s Rule - Math

1 Determinants and cramer s RuleUnique Solution of a2 2 SystemThe 2 2 systemax+by=e,cx+dy=f,(1)has a unique solution provided =ad bcis nonzero, in which case thesolution is given byx=de bfad bc, y=af cead bc.(2)This result, calledCramer s Rulefor 2 2 systems, is usually learnedin college algebra as part of determinant of Order2 College algebra introduces matrix notation and determinant notation:A=(a bc d),det(A) = a bc d .Evaluation of a 2 2 determinant is bySarrus Rule: abcd =ad boldface productadis the product of the main diagonal entries andthe other productbcis from the s 2 2 rule in determinant notation isx= e bf d a bc d , y= a ec f a bc d .(3)Unique Solution of ann nSystemCramer s rule can be generalized to ann nsystem of equationsA~x=~bora11x1+a12x2+ +a1nxn=b1,a21x1+a22x2+ +a2nxn=b2,..an1x1+an2x2+ +annxn=bn.(4) Determinants and cramer s Rule291 System (4) has a unique solution provided thedeterminant of coeffi-cients = det(A) is nonzero, in which case the solution is given byx1= 1 , x2= 2.

2 , xn= n .(5)The determinant jequals det(Bj) where matrixBjis matrixAwithcolumnjreplaced by~b= (b1,..,bn), which is the right side of system(4). The result is calledCramer s Ruleforn nsystems. Determinantswill be defined shortly; intuition from the 2 2 case and Sarrus ruleshould suffice for the Notation for cramer s coefficientsfor systemA~x=~bis denoted by = a11a12 a1na21a22 ..an1an2 ann .(6)The otherndeterminants in cramer s rule (5) are given by 1= b1a12 a1nb2a22 ..bnan2 ann , .. , n= a11a12 b1a21a22 ..an1an2 bn .(7)The literature is filled with conflicting notations for matrices, vectorsand Determinants . The reader should take care to use vertical barsonlyfor Determinants and absolute values, ,|A|makes sense for a matrixAor a constantA. For clarity, the notation det(A) is preferred, whenAis a matrix. The notation|A|implies thata determinant is a number,computed by|A|= Awhenn= 1, and|A|=a11a22 a12a21whenn=2.

3 Forn 3,|A|is computed by similar but increasingly complicatedformulas; see Sarrus rule and thefour Rule for3 algebra supplies thefollowing formula for the determinant of a 3 3 matrixA:det(A) = a11a12a13a21a22a23a31a32a33 =a11a22a33+a21a32a13+a31a12a23 a11a32a23 a21a12a33 a31a22a13.(8)292 The number det(A) can be computed by an algorithm similar to the onefor 2 2 matrices, as in Figure 8. We remark that no further generaliza-tions are possible:there is no Sarrus rule for4 4or larger matrices!a21a22a23a13a12a11a31a32a33a23a 22a21a11a12a13defabcFigure 8. Sarrus rule for3 3matrices,which givesdet(A) = (a+b+c) (d+e+f).College Algebra Definition of impracticaldefinition is the formuladet(A) = Sn( 1)parity( )a1 1 an n.(9)In the formula,aijdenotes the element in rowiand columnjof thematrixA. The symbol = ( 1,.., n) stands for a rearrangement ofthe subscripts 1, 2, .. ,nandSnis the set of all possible rearrange-ments.

4 The nonnegative integer parity( ) is determined by counting theminimum number of pairwise interchanges required to assemble the listof integers 1, .. , ninto natural order 1, .. , consequence of (9) is the relation det(A) = det(AT) whereATmeansthe transpose ofA, obtained by swapping rows and columns. This re-lation implies that all determinant theory results for rows also apply (9) reproduces the definition for 3 3 matrices given in equation(8). We will have no computational use for (9). For computing the valueof a determinant, see belowfour propertiesandcofactor definition of determinant (9) implies the fol-lowing four properties:TriangularThe value ofdet(A)for either an upper triangularor a lower triangular matrixAis the product of thediagonal elements:det(A) =a11a22 fromAby swapping two rows, thendet(A) = ( 1) det(B).CombinationThe value ofdet(A)is unchanged by adding a mul-tiple of a row to a different one row ofAis multiplied by constantcto creatematrixB, thendet(B) =cdet(A).

5 Determinants and cramer s Rule293It is known that these four rules suffice to compute the value of anyn ndeterminant. The proof of the four properties is delayed until page Matrices and the Four rules can bestated in terms of elementary matrices as value ofdet(A)for either an upper triangularor a lower triangular matrixAis the product of thediagonal elements:det(A) =a11a22 ann. Thisis a one-arrow Sarrus rule valid for an elementary matrix for a swap rule, thendet(EA) = ( 1) det(A).CombinationIfEis an elementary matrix for a combination rule,thendet(EA) = det(A).MultiplyIfEis an elementary matrix for a multiply rule withmultiplierc6= 0, thendet(EA) =cdet(A).Since det(E) = 1 for a combination rule, det(E) = 1 for a swap ruleand det(E) =cfor a multiply rule with multiplierc6= 0, it follows thatfor any elementary matrixEthere is the determinant multiplication ruledet(EA) = det(E) det(A).Additional Determinant following rules make for ef-ficient evaluation of certain special Determinants .

6 The results are statedfor rows, but they also hold for columns, because det(A) = det(AT).Zero rowIf one row ofAis zero, thendet(A) = rowsIf two rows ofAare identical, thendet(A) = (A)6=I, thendet(A) = factorThe relationdet(A) =cdet(B)holds, providedAandBdiffer only in one row, say rowj, for whichrow(A,j) =crow(B,j).Row linearityThe relationdet(A) = det(B) + det(C)holds, pro-videdA,BandCdiffer only in one row, say rowj, for whichrow(A,j) =row(B,j) +row(C,j).The proofs of these properties are delayed until page ExpansionThe special subject of cofactor expansions is used to justify cramer s ruleand to provide an alternative method for computation of is no claim that cofactor expansion is efficient, only that it ispossible, and different than Sarrus rule or the use of the four from College cofactor expansion the-ory is most easily understood from the college algebra topic, where thedimension is 3 and row expansion means the following formulas are valid:|A|= a11a12a13a21a22a23a31a32a33 =a11(+1) a22a23a32a33 +a12( 1) a21a23a31a33 +a13(+1) a21a22a31a32 =a21( 1) a12a13a32a33 +a22(+1) a11a13a31a33 +a23( 1) a11a12a31a32 =a31(+1) a12a13a22a23 +a32( 1) a11a13a21a23 +a33(+1) a11a12a21a22 The formulas expand a 3 3 determinant in terms of 2 2 Determinants ,along a row ofA.

7 The attached signs 1 are called thecheckerboardsigns, to be defined shortly. The 2 2 Determinants are calledminorsof the 3 3 determinant|A|. The checkerboard sign together with aminor is called formulas are generally used when a row has one or two zeros,making it unnecessary to evaluate one or two of the 2 2 determinantsin the expansion. To illustrate, row 1 expansion gives 3 0 02 1 75 4 8 = 3(+1) 1 74 8 = clever time saving choice is always a row which has the most zeros,although success does not depend upon cleverness. What has been saidfor rows also applies to columns, due to the transpose formula|A|=|AT|.Minors and (n 1) (n 1) determinant obtainedfrom det(A) by striking out rowiand columnjis called the (i,j) minorofAand denotedminor(A,i,j) (Mijis common in literature). The(i,j) cofactor ofAiscof(A,i,j) = ( 1)i+jminor(A,i,j). Multiplicativefactor ( 1)i+jis called thecheckerboard sign, because its value canbe determined by countingplus,minus,plus, etc.

8 , from location (1,1) tolocation (i,j) in any checkerboard of Determinants by formulas aredet(A) =n j=1akjcof(A,k,j),det(A) =n i=1ai`cof(A,i,`),(10) Determinants and cramer s Rule295where 1 k n, 1 ` n. The first expansion in (10) is calledacofactor row expansionand the second is called acofactor col-umn expansion. The valuecof(A,i,j) is the cofactor of elementaijindet(A), that is, the checkerboard sign times the minor ofaij. The proofof expansion (10) is delayed until page Adjugate (A) of ann nmatrixAis the transpose of the matrix of cofactors,adj(A) = cof(A,1,1)cof(A,1,2) cof(A,1,n)cof(A,2,1)cof(A,2,2) cof(A,2,n)..cof(A,n,1)cof(A,n,2) cof(A,n,n) cofactorcof(A,i,j) is the checkerboard sign ( 1)i+jtimes the corre-sponding minor determinantminor(A,i,j). In the 2 2 case,adj(a11a12a21a22)=(a22 a12 a21a11)In words:swap the diagonalelements and change the signof the off diagonal Inverse adjugate appears in the formula for theinverse matrixA 1:(a11a12a21a22) 1=1a11a22 a12a21(a22 a12 a21a11).

9 This formula is verified by direct matrix multiplication:(a11a12a21a22) (a22 a12 a21a11)= (a11a22 a12a21)(1 00 1).For ann nmatrix,A adj(A) = det(A)I, which gives the formulaA 1=1det(A) cof(A,1,1)cof(A,1,2) cof(A,1,n)cof(A,2,1)cof(A,2,2) cof(A,2,n)..cof(A,n,1)cof(A,n,2) cof(A,n,n) TThe proof ofA adj(A) = det(A)Iis delayed to page elementary matrixEis the result ofapplying a combination, multiply or swap rule to the identity definition implies that an elementary matrix is the identity matrixwith a minor change applied, to wit:296 CombinationChange an off-diagonal zero a diagonal one ofIto multiplierm6= two rows 9 ( Determinants and elementary Matrices)LetEbe ann nelementary matrix. ThenCombinationdet(E) = 1 Multiplydet(E) =mfor (E) = 1 Productdet(EX) = det(E) det(X)for alln 10 ( Determinants and Invertible Matrices)LetAbe a given invertible matrix. Thendet(A) =( 1)sm1m2 mrwheresis the number of swap rules applied andm1,m2.

10 ,mrare thenonzero multipliers used in multiply rules whenAis reduced torref(A).Determinant Product determinant rules of combina-tion, multiply and swap imply that det(EX) = det(E) det(X) for el-ementary matricesEand square matricesX. We show that a moregeneral relationship 11 (Determinant Product Rule)LetAandBbe givenn nmatrices. Thendet(AB) = det(A) det(B).Proof:Used in the proof is the equivalence of invertibility of a square matrixCwithdet(C)6= 0 andrref(C) = one ofAorBhas zero determinant. Then det(A) det(B) = 0. Ifdet(B) = 0, thenBx=0has infinitely many solutions, in particular a nonzerosolutionx. MultiplyBx=0byA, thenABx=0which impliesABis notinvertible. Then the identity det(AB) = det(A) det(B) holds, because bothsides are zero. If det(B)6= 0 but det(A) = 0, then there is a nonzeroywithAy=0. Definex=AB 1y. ThenABx=Ay=0, withx6=0, whichimplies the identity This completes the proof when one ofAorBis ,Bare invertible and thenC=ABis invertible.