Linear AlgebraDeterminantsThe 2 x 2 case Consider the system of linear equations which can be written in matrix form as This system has a solution if and only if ad - bc ≠ 0. Thenumber ad - bc, which also appears in the solution of theabove equations, is called the determinant of the coefficientmatrix.1 2 11 2 2ax bx ycx dx y1122xyabxycdGeneralizing to a square matrix of any size: Let A = [ aij] be an nxn matrix. The determinant of A is defined aswhere the summation is over all permutations j1 j2… jnofS = { 1, 2, …, n }. The sign is positive if j1 j2… jnis aneven permutation and is negative if j1 j2… jnis an oddpermutation .1212detnnjjja a aA Permutations Let S = { 1, 2, …, n } be the integers 1 through n, arranged in ascending order. A rearrangement j1 j2… jnof the elements of S is called a permutation of S. A permutation of S is a one to one mapping of S onto itself. The number of permutations of S = { 1, 2, …, n } is n! The set of all permutations of the elements of S is denoted by SnInversions A permutation j1 j2… jnof S is said to have an inversion if a larger integer, jr, precedes a smaller one, js.  Example - Let S = { 1, 2, 3 }. Then S3= { 123, 132, 213, 231, 312, 321 }123 - 0 inversions132 - 1 inversion213 - 1 inversion231 - 2 inversions312 - 2 inversions321 - 3 inversionsEven and Odd Permutations A permutation is called even if the total number of inversions is even. A permutation is called odd if the total number of inversions is odd. 4312 contains 5 inversions and is an odd permutation.  1423 contains 2 inversions and is an even permutation.Calculating the Determinant Each term of det ( A ) has row subscripts in natural order and column subscripts in the order j1 j2… jn. Each term is a product of n elements of A, with exactly one entry from each row of Aand exactly one entry from each column of A. det ( A ) may be written1212nnjjja a aAThe 3 x 3 Case Letdet (A) = a11 a22 a33+ a12 a23 a31 + a13 a21 a32-a11 a23 a32 - a12 a21 a33 -a13 a22 a3111 12 1321 22 2331 32 33a a aa a aa a aAProperties of Determinants Let A be a square matrix:  det ( A ) = det ( AT)  If matrix B is obtained from matrix A by interchanging two rows (columns) of A, then det ( B ) = - det ( A ). If two rows (columns) of A are equal, then det ( A ) = 0. If a row (column) of A consists entirely of zeros, then det ( A ) = 0.More Properties of Determinants Let A be a square matrix: If a row (column) of A consists entirely of zeros, then det ( A ) = 0. If B is obtained from A by multiplying a row (column) of A by a real number c, then det( B ) = c det ( A ).  If B = [ bij] is obtained from A = [ aij] by adding to each element of r th row (column) of A, c times the corresponding element of the s th row (column), r ≠ s, of A, then det( B ) =det( A ).Yet More Properties Let A = [ aij] be an upper (lower) triangular matrix, then det( A ) = a11a22… ann. That is, the determinant of a triangular matrix is just the product of the elements on the main diagonal. Let A be a n x n matrix. Then A is nonsingular if and only if det( A ) ≠ 0. If A and B are n x n matrices, then det( AB ) = det( A ) det( B )How many of the properties you could prove if asked?A Short Proof If A is an n x n matrix, then Ax = 0 has a nontrivial solution if and only if det( A ) = 0.Proof:(i) If Ax = 0 has a nontrivial solution, then Ais singular and det( A ) = 0.(ii) If det( A ) = 0, then A is singular and Ax = 0 has a nontrivial solution. QEDA Short Proof If A is nonsingular, then det( A–1) = 1 / det( A ) Proof:(i) Let Inbe the n x n identity matrix.(ii) AA–1= Inand det( In) = 1.(iii) By theorem, det (AB) = det(A)det(B), so since (iv) det(AA–1) = det( In) = 1 det( A ) det( A–1) =1 det( A–1) = 1 / det( A ).QEDCofactor ExpansionLet1 2 34 5 67 8 9A11 12 1321 22 2331 32 335 6 4 6 4 548 42 38 0 7 0 7 82 3 1 3 1 224 21 68 0 7 0 7 82 3 1 3 1 23 6 35 6 4 6 4 5A A AA A AA A ACofactor Example (Continued)11 21 3112 22 3213 23 3348 24 3adj 42 21 63 6 3A A AA A AA A AA31 32det 7 8 7 3 8 6 27AAA1 2 3 48 24 3 27 0 04 5 6 42 21 6 0 27 07 8 0 3 6 3 0 0 27Theorems Let A = [ aij] be an n x n matrix. Then A ( adj( A ) ) = ( adj( A ) ) A = det( A ) In.Theorem Let A be an n x n matrix with det( A ) ≠ 0. Then111 21212 22112det det det1det det detadjdetdet det detnnn n nnAAAAAAAAAA A AA A AAAAA A A   Proof of Previous Theorem By a preceding theorem, A ( adj( A ) ) = det( A ) In., Then adj( A ) = det( A ) A–1and 11adjdetAAAInverses by Determinants1 1 10 2 35 5 1A8A11 12 132 3 0 3 0 2 5 1 5 1 5 5M M M21 22 231 1 1 1 1 1 5 1 5 1 5 5M M M31 32 331 1 1 1 1 1 2 3 0 3 0 3M M MInverses by Determinants (cont’d)13 15 10 4 4 0 1 3 2C13 4 1 15 4 3 -10 0 2TC111det( ) 8TACA13 4 1 15 4 3 -10 0 213 1 1 8 2 815 1 3 8 2 851 044Cramer’s Rule Consider the linear system of n equations in nunknownsLet A = [ aij] be the coefficient matrix, so that thesystem may be expressed as Ax = b. If det( A ) ≠ 0, then the system has a uniquesolution. 11 1 12 2 1 121 1 22 2 2 21 1 2 2nnnnnn n nnna x a x a x ba x a x a x ba x a x a x b   Cramer’s Rule The solution has the form1212det detdet, , ,det det detnnx x xAAAA A AWhere Ai is the matrix obtained from A by replacing its i th column with b.Example:1 3 72 1 …