Determinants PDF - March 2022

Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule DETERMINANTS Dr. Gabriel Obed Fosu Department of Mathematics Kwame Nkrumah University of Science and Technology...

Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule DETERMINANTS Dr. Gabriel Obed Fosu Department of Mathematics Kwame Nkrumah University of Science and Technology March 28, 2022 Dr. Gabriel Obed Fosu 1/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Outline 1 Developing the Determinant of a Matrix Introduction Determinant of n × n Matrix Cofactors, Adjoint, and Inverse of a Matrix 2 Some Properties of Determinant 3 Cramer’s Rule Dr. Gabriel Obed Fosu 2/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Outline of Presentation 1 Developing the Determinant of a Matrix Introduction Determinant of n × n Matrix Cofactors, Adjoint, and Inverse of a Matrix 2 Some Properties of Determinant 3 Cramer’s Rule Dr. Gabriel Obed Fosu 3/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Introduction Definition The determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Dr. Gabriel Obed Fosu 4/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Introduction Definition The determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. For example, in terms of linear algebra, determinants can be used to 1 characterize nonsingular matrices, 2 Used to express solutions of nonsingular systems Ax = b 3 Used to express vector cross products. Dr. Gabriel Obed Fosu 4/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Determinant of Matrix Theorem Let Mij denote the determinant of the (n − 1) × (n − 1) submatrix of A formed by deleting the ith row and jth column of A. Assume that the determinant function has been defined for matrices of size (n − 1) × (n − 1). Then the determinant of the n × n matrix A is defined by what we call the first-row Laplace expansion: n X |A| = (−1)1+j a1j M1j (1) j=1 = a11 M11 − a12 M12 + · · · + (−1)1+n M1n (2) The values Mij are termed minors. Dr. Gabriel Obed Fosu 5/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Determinant of 2 × 2 Matrix 1 The determinant of the general 2 × 2 matrix   a a12 A = 11 a21 a22 is |A| = a11 (a22 ) − a12 (a21 ) 2 The minors are a22 and a21. Dr. Gabriel Obed Fosu 6/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Determinant of 2 × 2 Matrix 1 The determinant of the general 2 × 2 matrix   a a12 A = 11 a21 a22 is |A| = a11 (a22 ) − a12 (a21 ) 2 The minors are a22 and a21. Example  11 2 The determinant of Z = is 3 −1 |Z| = 11(−1) − 2(3) = −17 (3) Dr. Gabriel Obed Fosu 6/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Determinant of 3 × 3 Matrix Let   a11 a12 a13 A = a21 a22 a23  a31 a32 a33 then the determinant is defined as a22 a23 |A| = a11 a32 a33 Dr. Gabriel Obed Fosu 7/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Determinant of 3 × 3 Matrix Let   a11 a12 a13 A = a21 a22 a23  a31 a32 a33 then the determinant is defined as a22 a23 a a23 |A| = a11 − a12 21 a32 a33 a31 a33 Dr. Gabriel Obed Fosu 7/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Determinant of 3 × 3 Matrix Let   a11 a12 a13 A = a21 a22 a23  a31 a32 a33 then the determinant is defined as a22 a23 a a23 a a22 |A| = a11 − a12 21 + a13 21 (4) a32 a33 a31 a33 a31 a32 Dr. Gabriel Obed Fosu 7/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Determinant of 3 × 3 Matrix Let   a11 a12 a13 A = a21 a22 a23  a31 a32 a33 then the determinant is defined as a22 a23 a a23 a a22 |A| = a11 − a12 21 + a13 21 (4) a32 a33 a31 a33 a31 a32 = a11 [a22 (a33 ) − a23 a32 ] − a12 [a21 (a33 ) − a23 (a31 )] + a13 [a21 (a32 ) − a22 (a31 )] (5) Dr. Gabriel Obed Fosu 7/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Example   1 2 −1 Compute the determinant of the 3 × 3 matrix C = 0 4 1 3 5 −9 Dr. Gabriel Obed Fosu 8/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Example   1 2 −1 Compute the determinant of the 3 × 3 matrix C = 0 4 1 3 5 −9 4 1 |C| =1 5 −9 Dr. Gabriel Obed Fosu 8/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Example   1 2 −1 Compute the determinant of the 3 × 3 matrix C = 0 4 1 3 5 −9 4 1 0 1 |C| =1 −2 5 −9 3 −9 Dr. Gabriel Obed Fosu 8/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Example   1 2 −1 Compute the determinant of the 3 × 3 matrix C = 0 4 1 3 5 −9 4 1 0 1 0 4 |C| =1 −2 + (−1) (6) 5 −9 3 −9 3 5 Dr. Gabriel Obed Fosu 8/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Example   1 2 −1 Compute the determinant of the 3 × 3 matrix C = 0 4 1 3 5 −9 4 1 0 1 0 4 |C| =1 −2 + (−1) (6) 5 −9 3 −9 3 5 = 1[4(−9) − 1(5)] − 2[0(−9) − 1(3)] − 1[0(5) − 3(4)] (7) = 1(−41) − 2(−3) − 1(−12) (8) = −23 (9) Dr. Gabriel Obed Fosu 8/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix The evaluation of an n × n matrix was presented in terms of the first-row expansion. Actually, we can expand the determinant along any row or column 1 The ith row expansion is n X |A| = (−1)i+j aij Mij (10) j=1 Dr. Gabriel Obed Fosu 9/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix The evaluation of an n × n matrix was presented in terms of the first-row expansion. Actually, we can expand the determinant along any row or column 1 The ith row expansion is n X |A| = (−1)i+j aij Mij (10) j=1 2 The jth column expansion is n X |A| = (−1)i+j aij Mij (11) i=1 Dr. Gabriel Obed Fosu 9/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix The evaluation of an n × n matrix was presented in terms of the first-row expansion. Actually, we can expand the determinant along any row or column 1 The ith row expansion is n X |A| = (−1)i+j aij Mij (10) j=1 2 The jth column expansion is n X |A| = (−1)i+j aij Mij (11) i=1 3 The expression (−1)i+j obeys the chessboard pattern of signs:   + − + ··· − + − · · ·    + − + · · · ........  .... Dr. Gabriel Obed Fosu 9/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Cofactors, Adjoint and Inverse of Matrices Definition (Cofactor) The (i, j) cofactor of A, denoted by Cij is defined by Cij = (−1)i+j Mij (12) Dr. Gabriel Obed Fosu 10/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Cofactors, Adjoint and Inverse of Matrices Definition (Cofactor) The (i, j) cofactor of A, denoted by Cij is defined by Cij = (−1)i+j Mij (12) Definition (Adjoint) If A = [aij ] is an n × n matrix, the adjoint of A, denoted by adj A, is the transpose of the matrix of cofactors. Dr. Gabriel Obed Fosu 10/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Cofactors, Adjoint and Inverse of Matrices Definition (Cofactor) The (i, j) cofactor of A, denoted by Cij is defined by Cij = (−1)i+j Mij (12) Definition (Adjoint) If A = [aij ] is an n × n matrix, the adjoint of A, denoted by adj A, is the transpose of the matrix of cofactors. Definition (Inverse) The inverse of a matrix is given by the relation 1 A−1 = adj A det A Dr. Gabriel Obed Fosu 10/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Example   1 2 3 Find the inverse of A = 4 5 6 8 8 9 Dr. Gabriel Obed Fosu 11/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Example   1 2 3 Find the inverse of A = 4 5 6 8 8 9 1 The determinant is 5 6 4 6 4 5 |A| = 1 −2 +3 8 9 8 9 8 8 Dr. Gabriel Obed Fosu 11/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Example   1 2 3 Find the inverse of A = 4 5 6 8 8 9 1 The determinant is 5 6 4 6 4 5 |A| = 1 −2 +3 = −3 + 24 − 24 = −3 (13) 8 9 8 9 8 8 Dr. Gabriel Obed Fosu 11/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Example   1 2 3 Find the inverse of A = 4 5 6 8 8 9 1 The determinant is 5 6 4 6 4 5 |A| = 1 −2 +3 = −3 + 24 − 24 = −3 (13) 8 9 8 9 8 8 2 The cofactor matrix is   5 6 4 6 4 5  8 −  9 8 9 8 8    2 3 1 3 1 2  − −   8 9 8 9 8 8     2 3 1 3 1 2  − 5 6 4 6 4 5 Dr. Gabriel Obed Fosu 11/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix Example   1 2 3 Find the inverse of A = 4 5 6 8 8 9 1 The determinant is 5 6 4 6 4 5 |A| = 1 −2 +3 = −3 + 24 − 24 = −3 (13) 8 9 8 9 8 8 2 The cofactor matrix is   5 6 4 6 4 5  8 − 9 8 9 8 8      2  −3 12 −8 3 1 3 1 2  −  8 −  = 6 −15 8  9 8 9 8 8    2  −3 6 −3 3 1 3 1 2  − 5 6 4 6 4 5 Dr. Gabriel Obed Fosu 11/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix 1 The adjoint is given by   −3 6 −3 adj A =  12 −15 6  −8 8 −3 Dr. Gabriel Obed Fosu 12/36  Developing the Determinant of a Matrix Introduction Some Properties of Determinant Determinant of n × n Matrix Cramer’s Rule Cofactors, Adjoint, and Inverse of a Matrix 1 The adjoint is given by   −3 6 −3 adj A =  12 −15 6  −8 8 −3 2 Inverse   −3 6 −3 1 A−1 =− 12 −15 6  (14) 3 −8 8 −3   1 −2 1 =  −4 5 −2 (15) 8/3 −8/3 1 Dr. Gabriel Obed Fosu 12/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Outline of Presentation 1 Developing the Determinant of a Matrix Introduction Determinant of n × n Matrix Cofactors, Adjoint, and Inverse of a Matrix 2 Some Properties of Determinant 3 Cramer’s Rule Dr. Gabriel Obed Fosu 13/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Definition The product of a matrix A and its adjoint is a diagonal matrix whose diagonal entries are det(A).     1 2 3 −3 6 −3 For a matrix A = 4 5 6 it adjoint its adj A =  12 −15 6  then 8 8 9 −8 8 −3    1 2 3 −3 6 −3 A × adjA = 4 5 6  12 −15 6  (16) 8 8 9 −8 8 −3   −3 0 0 =  0 −3 0  (17) 0 0 −3 Dr. Gabriel Obed Fosu 14/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Definition The product of a matrix A and its adjoint is a diagonal matrix whose diagonal entries are det(A).     1 2 3 −3 6 −3 For a matrix A = 4 5 6 it adjoint its adj A =  12 −15 6  then 8 8 9 −8 8 −3    1 2 3 −3 6 −3 A × adjA = 4 5 6  12 −15 6  (16) 8 8 9 −8 8 −3   −3 0 0 =  0 −3 0  (17) 0 0 −3 Note 5 6 4 6 4 5 |A| = 1 −2 +3 = −3 + 24 − 24 = −3 (18) 8 9 8 9 8 8 Dr. Gabriel Obed Fosu 14/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinant Definition The determinant of a lower triangular matrix   a11 0 ··· 0  a21 a22 ··· 0  ........   ....   an1 an2 ··· ann is the product of the diagonal elements a11 × a22 × a33 × · · · × ann The same result applies to the determinant of an upper triangular matrix and a diagonal Matrix Dr. Gabriel Obed Fosu 15/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example   1 2 −1 Compute the determinant of the 3 × 3 matrix C = 0 1 2 0 0 −9 Dr. Gabriel Obed Fosu 16/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example   1 2 −1 Compute the determinant of the 3 × 3 matrix C = 0 1 2 0 0 −9 Expanding by the second column, we have |C| = 1[1(−9) − 0(2)] + 2[0(−9) − 0(2)] − 1[1(0) − 0(0)] (19) = −9 + 0 + 0 (20) = −9 (21) Dr. Gabriel Obed Fosu 16/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example   1 2 −1 Compute the determinant of the 3 × 3 matrix C = 0 1 2 0 0 −9 Expanding by the second column, we have |C| = 1[1(−9) − 0(2)] + 2[0(−9) − 0(2)] − 1[1(0) − 0(0)] (19) = −9 + 0 + 0 (20) = −9 (21) or |C| = a11 × a22 × a33 = 1 × 1 × −9 = −9 (22) Dr. Gabriel Obed Fosu 16/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinant Definition A matrix and its transpose have equal determinants; that is |A| = |AT | Dr. Gabriel Obed Fosu 17/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinant Definition A matrix and its transpose have equal determinants; that is |A| = |AT | If   1 2 −1 C = 0 1 2 0 0 −9 then   1 0 0 CT =  2 1 0 (23) −1 2 −9 Thus |C T | = a11 × a22 × a33 = 1 × 1 × −9 = −9 (24) Dr. Gabriel Obed Fosu 17/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinant Definition If a row or column of a matrix is zero, then the value of the determinant is 0. Dr. Gabriel Obed Fosu 18/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinant Definition If a row or column of a matrix is zero, then the value of the determinant is 0. Example   1 2 −1 Compute the determinant of the 3 × 3 matrix C = 0 0 0 3 5 −9 Dr. Gabriel Obed Fosu 18/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinant Definition If a row or column of a matrix is zero, then the value of the determinant is 0. Example   1 2 −1 Compute the determinant of the 3 × 3 matrix C = 0 0 0 3 5 −9 Expanding by the second column, we have |C| = −0[2(−9) − 1(5)] + 0[1(−9) − 1(3)] − 0[1(5) − 3(2)] (25) =0+0+0 (26) =0 (27) Dr. Gabriel Obed Fosu 18/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Definition For an n × n matrix A det(cA) = cn det(A); where c is any scalar (28) Dr. Gabriel Obed Fosu 19/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Definition For an n × n matrix A det(cA) = cn det(A); where c is any scalar (28)     1 0 0 0 10 0 0 0 0 1 0 0  0 10 0 0 If A =  0  and c = 10 then cA =   and such 0 1 0 0 0 10 0  0 0 0 1 0 0 0 10 Dr. Gabriel Obed Fosu 19/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Definition For an n × n matrix A det(cA) = cn det(A); where c is any scalar (28)     1 0 0 0 10 0 0 0 0 1 0 0  0 10 0 0 If A =  0  and c = 10 then cA =   and such 0 1 0 0 0 10 0  0 0 0 1 0 0 0 10 det(cA) = 10 × 10 × 10 × 10 = 104 (29) Dr. Gabriel Obed Fosu 19/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Definition For an n × n matrix A det(cA) = cn det(A); where c is any scalar (28)     1 0 0 0 10 0 0 0 0 1 0 0  0 10 0 0 If A =  0  and c = 10 then cA =   and such 0 1 0 0 0 10 0  0 0 0 1 0 0 0 10 det(cA) = 10 × 10 × 10 × 10 = 104 (29) and cn det(A) = 104 (1 × 1 × 1 × 1) = 104 (30) Dr. Gabriel Obed Fosu 19/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Definition The determinants of det(AB) = (det A)(det B) (31) Dr. Gabriel Obed Fosu 20/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Definition The determinants of det(AB) = (det A)(det B) (31)       1 0 0 0 10 0 0 0 10 0 0 0 0 1 0 0 If A =   and B =  0 10 0 0  then AB =  0 10 0 0     0 0 1 0 0 0 10 0  0 0 10 0 0 0 0 1 0 0 0 10 0 0 0 10 Dr. Gabriel Obed Fosu 20/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Definition The determinants of det(AB) = (det A)(det B) (31)       1 0 0 0 10 0 0 0 10 0 0 0 0 1 0 0 If A =   and B =  0 10 0 0  then AB =  0 10 0 0 Thus    0 0 1 0 0 0 10 0  0 0 10 0 0 0 0 1 0 0 0 10 0 0 0 10 det(AB) = 10 × 10 × 10 × 10 = 104 (32) Dr. Gabriel Obed Fosu 20/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Definition The determinants of det(AB) = (det A)(det B) (31)       1 0 0 0 10 0 0 0 10 0 0 0 0 1 0 0 If A =   and B =  0 10 0 0  then AB =  0 10 0 0 Thus    0 0 1 0 0 0 10 0  0 0 10 0 0 0 0 1 0 0 0 10 0 0 0 10 det(AB) = 10 × 10 × 10 × 10 = 104 (32) and det(A) det(B) = (1 × 1 × 1 × 1)(10 × 10 × 10 × 10) = 104 (33) Dr. Gabriel Obed Fosu 20/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants A determinant is a linear function of each row separately. If two rows are added, with all other rows remaining the same, the determinants are added. Dr. Gabriel Obed Fosu 21/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants A determinant is a linear function of each row separately. If two rows are added, with all other rows remaining the same, the determinants are added. 2 3 4 −1 −2 −3 = 2 (34) −4 −3 −4 5 6 7 −1 −2 −3 = 8 (35) −4 −3 −4   2+5 3+6 4+7 7 9 11 −1 −2 −3 = −1 −2 −3 = 10 (36) −4 −3 −4 −4 −3 −4 Dr. Gabriel Obed Fosu 21/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Scalar Multiplication If a row of A is multiplied by a scalar t, then the determinant of the modified matrix is t det A. Dr. Gabriel Obed Fosu 22/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Scalar Multiplication If a row of A is multiplied by a scalar t, then the determinant of the modified matrix is t det A. 1 4 0 2 5 1 =4 (37) 1 0 0 Then multiplying row 2 by 7, we have 1 4 0 14 35 7 = 28 = 4(7) (38) 1 0 0 Dr. Gabriel Obed Fosu 22/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Row Swap If two rows of a matrix are exchanged, the determinant changes sign. Dr. Gabriel Obed Fosu 23/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Row Swap If two rows of a matrix are exchanged, the determinant changes sign. 1 0 0 2 5 1 = −4 (39) 1 4 0 Interchanging rows 2 and 3 1 0 0 1 4 0 =4 (40) 2 5 1 Dr. Gabriel Obed Fosu 23/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Row Arithmetic If a multiple of a row is subtracted from another row, the value of the determinant remains unchanged. Dr. Gabriel Obed Fosu 24/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Row Arithmetic If a multiple of a row is subtracted from another row, the value of the determinant remains unchanged. 1 0 0 1 0 0 −→ 1 4 0 1 4 0 =4 (41) N R3 = R3 − 8R2 2 5 1 −6 −27 1 Dr. Gabriel Obed Fosu 24/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Row Arithmetic If a multiple of a row is subtracted from another row, the value of the determinant remains unchanged. 1 0 0 1 0 0 −→ 1 4 0 1 4 0 =4 (41) N R3 = R3 − 8R2 2 5 1 −6 −27 1 Equal Rows When two rows of a matrix are equal, the determinant is zero. Dr. Gabriel Obed Fosu 24/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Properties of Determinants Row Arithmetic If a multiple of a row is subtracted from another row, the value of the determinant remains unchanged. 1 0 0 1 0 0 −→ 1 4 0 1 4 0 =4 (41) N R3 = R3 − 8R2 2 5 1 −6 −27 1 Equal Rows When two rows of a matrix are equal, the determinant is zero. 1 0 1 2 1 8 =0 (42) 1 0 1 Dr. Gabriel Obed Fosu 24/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Theorem 1 A matrix A is nonsingular if and only if det A ̸= 0 2 A is singular if and only if det A = 0 3 The homogeneous system Ax = 0 has a nontrivial solution if and only if det A = 0. Dr. Gabriel Obed Fosu 25/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Theorem 1 A matrix A is nonsingular if and only if det A ̸= 0 2 A is singular if and only if det A = 0 3 The homogeneous system Ax = 0 has a nontrivial solution if and only if det A = 0. Example Find numbers a for which the following homogeneous system has a nontrivial solution and solve the system for these values of a: x − 2y + 3z = 0, ax + 3y + 2z = 0, 6x + y + az = 0 Dr. Gabriel Obed Fosu 25/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule   1 −2 3 A coefficient matrix say D = a 3 2 6 1 a Then |D| = (3a − 2) + 2(a2 − 12) + 3(a − 18) (43) 2 = 2a + 6a − 80 (44) This |D| = 0 =⇒ 2a2 + 6a − 80 = 0 =⇒ a = −8 or a = 5 These values of a are the only values for which the given homogeneous system has a nontrivial solution. Dr. Gabriel Obed Fosu 26/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule   1 −2 3 1 When a = −8 we obtain D = −8 3 2 6 1 −8 2 ERO will lead to the upper triangular (that is the augmented matrix)   1 0 −1 0 0 1 −2 0 (45) 0 0 0 0 3 Solving this system gives the nontrivial solution x = z and y = 2z Exercise The case of a = 5 is left as an exercise. Dr. Gabriel Obed Fosu 27/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Outline of Presentation 1 Developing the Determinant of a Matrix Introduction Determinant of n × n Matrix Cofactors, Adjoint, and Inverse of a Matrix 2 Some Properties of Determinant 3 Cramer’s Rule Dr. Gabriel Obed Fosu 28/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Cramer’s Rule Definition (Cramer’s Rule) Let A be an n × n invertible matrix, and let b be a column vector with n components. Let Ai be the  matrix obtained by replacing the ith column of A with b. x1  x2  · · · is the unique solution to the linear system Ax = b, then If x =   xn det(Ai ) xi = ; i = 1, 2, · · · , n (46) det(A) Dr. Gabriel Obed Fosu 29/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example Use Cramer’s rule to solve the linear system. 2x + 3y = 2 (47) −5x + 7y = 3 (48) Dr. Gabriel Obed Fosu 30/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example Use Cramer’s rule to solve the linear system. 2x + 3y = 2 (47) −5x + 7y = 3 (48) The determinant of the coefficient matrix is given by 2 3 = 14 − (−15) = 29 (49) −5 7 Dr. Gabriel Obed Fosu 30/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example Use Cramer’s rule to solve the linear system. 2x + 3y = 2 (47) −5x + 7y = 3 (48) The determinant of the coefficient matrix is given by 2 3 = 14 − (−15) = 29 (49) −5 7 and since the determinant is not zero, the system has a unique solution given by 2 3 3 7 14 − 9 5 x= = = (50) 29 29 29 Dr. Gabriel Obed Fosu 30/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example Use Cramer’s rule to solve the linear system. 2x + 3y = 2 (47) −5x + 7y = 3 (48) The determinant of the coefficient matrix is given by 2 3 = 14 − (−15) = 29 (49) −5 7 and since the determinant is not zero, the system has a unique solution given by 2 3 3 7 14 − 9 5 x= = = (50) 29 29 29 2 2 −5 3 6 − (−10) 16 y= = = (51) 29 29 29 Dr. Gabriel Obed Fosu 30/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example Solve the linear system 2x + 3y − z = 2 (52) 3x − 2y + z = −1 (53) −5x − 4y + 2z = 3 (54) (55) Dr. Gabriel Obed Fosu 31/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example Solve the linear system 2x + 3y − z = 2 (52) 3x − 2y + z = −1 (53) −5x − 4y + 2z = 3 (54) (55) The determinant of the coefficient matrix is given by 2 3 −1 3 −2 1 = −11 (56) −5 −4 2 Dr. Gabriel Obed Fosu 31/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule By Cramer’s rule the solution to the system is 2 3 −1 1 5 x=− −1 −2 1 = − (57) 11 11 3 −4 2 Dr. Gabriel Obed Fosu 32/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule By Cramer’s rule the solution to the system is 2 3 −1 1 5 x=− −1 −2 1 = − (57) 11 11 3 −4 2 2 2 −1 1 36 y=− 3 −1 1 =− (58) 11 11 −5 3 2 Dr. Gabriel Obed Fosu 32/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule By Cramer’s rule the solution to the system is 2 3 −1 1 5 x=− −1 −2 1 = − (57) 11 11 3 −4 2 2 2 −1 1 36 y=− 3 −1 1 =− (58) 11 11 −5 3 2 2 3 2 1 76 z=− 3 −2 −1 = − (59) 11 11 −5 −4 3 Dr. Gabriel Obed Fosu 32/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Alternative Method for Solving Determinant Another method for calculating the determinant of a 3 × 3 matrix A is follows 1 Copy the first two columns of A to the right of the matrix 2 Take the products of the elements on the six diagonals shown below. 3 Attach plus signs to the products from the downward-sloping diagonals 4 Attach minus signs to the products from the upward-sloping diagonals. Dr. Gabriel Obed Fosu 33/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Alternative Method for Solving Determinant Another method for calculating the determinant of a 3 × 3 matrix A is follows 1 Copy the first two columns of A to the right of the matrix 2 Take the products of the elements on the six diagonals shown below. 3 Attach plus signs to the products from the downward-sloping diagonals 4 Attach minus signs to the products from the upward-sloping diagonals. |A| = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 − a31 a22 a13 − a32 a23 a11 − a33 a21 a12 Dr. Gabriel Obed Fosu 33/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example   5 −3 2 Calculate the determinant of the matrix A = 1 0 2 2 −1 3 Dr. Gabriel Obed Fosu 34/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example   5 −3 2 Calculate the determinant of the matrix A = 1 0 2 2 −1 3 We adjoin to A its first two columns and compute the six indicated products: Dr. Gabriel Obed Fosu 34/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Example   5 −3 2 Calculate the determinant of the matrix A = 1 0 2 2 −1 3 We adjoin to A its first two columns and compute the six indicated products: Adding the three products at the bottom and subtracting the three products at the top gives det(A) = 0 + (−12) + (−2) − 0 − (−10) − (−9) = 5 (60) Dr. Gabriel Obed Fosu 34/36  Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule Exercises     1 −1 0 0 1 0 −2 −3 −3 −1 −1 1 Given the matrix A = 3 1 4  and B =  −1  Compute −1 −3 2 5 2 −3 −1 −2 2 1 a. The determinant b. The cofactor matrix c. The inverse 2 Show that 1 1 1 1 r 1 1 1 = (1 − r)3 r r 1 1 r r r 1 Dr. Gabriel Obed Fosu 35/36 Developing the Determinant of a Matrix Some Properties of Determinant Cramer’s Rule END OF LECTURE THANK YOU Dr. Gabriel Obed Fosu 36/36