MATH 1140(M12A) Semester I. Ax =0 (1) x 4. x 2. x 3. By using Gaussian Elimination, we obtain the solution

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1 MATH 4(M2A) Semester I Homogeneous Systems These are systems of linear equations of the form Ax = () whereais the coefficient matrix and x is the vector of unknowns. Example x 2x 2 4x 3 3x 4 = 2x +x 2 3x 3 x 4 = 2x +3x 2 x 3 +x 4 = 3x +5x 2 x 3 +2x 4 = By using Gaussian Elimination, we obtain the solution x x 2 x 3 x 4 =s 2 +t Homogeneous systems are always consistent, since the zero vector x = is always a solution to (). Therefore, a homogeneous system either has a unique solution x = or an infinite number of solutions. Lemma 2 If z is a solution of () then λz is also a solution of () for any scalar λ. If z and z 2 are solutions of () then z +z 2 is also a solution of ()..

2 These facts follow from the properties of matrix multiplication. A(λz ) =λ(az ) =λ =. A(z +z 2 ) =Az +Az 2 =+ =. The general solution of () can always be written in the form λ z +λ 2 z λ k z k, wherez, z 2,...,z k are vectors of solutions andλ,λ 2,...,λ k are parameters representing scalars. The general solution gives all solutions of the homogeneous system when all possible values are assigned to the parameters. When Gaussian Elimination is used to solve the system, the parametersλ,λ 2,..., λ k arise from the free variables. For the non-homogeneous system Ax =c (2) let x be a particular solution, so that If x is any solution of (2), then Ax =c. A(x x ) =Ax Ax =c c =. Thus, (x x ) =z is a solution of the homogeneous system (), i.e. x =x +z where z is a solution of (). Conversely, if z is a solution of the homogeneous system (), then A(x +z) =Ax +Az =c+ =c. Thus, x =x +z is a solution of the non-homogeneous system (2). Therefore we can write the general solution of the non-homogeneous system (2) as x =x +z where x is a particular solution of (2) and z is the general solution of the homogeneous system (). 2

3 Example 3 Consider the system x 2x 2 4x 3 3x 4 = 5 (3) 2x +x 2 3x 3 x 4 = 5 2x +3x 2 x 3 +x 4 = 3x +5x 2 x 3 +2x 4 = 8 We can easily verify that x x 2 x 3 x 4 = 3 is a particular solution of the system. Also, x x 2 x 3 x 4 =s 2 +t is the general solution of the corresponding homogeneous system as we showed in Example. Therefore, the general solution of the system (3) is x x 2 x 3 x 4 = 3 +s 2 +t. 3

4 2 Definition of Determinant Definition 4 Let A be an n n matrix, where n 2. The ij th minor matrix ofa, denoted bym ij is the matrix obtained by removing theith row andj th column ofa. Definition 5 IfAis ann n matrix, where n 2, then the determinant ofa, denoted by A or det A, is defined by A =a M a 2 M 2 +a 3 M ( ) n+ a n M n (4) wherem ij is theij th minor matrix ofa. The determinant of a matrix, A = (a ), is A =a. The expression (4) is called theexpansionofthedeterminant ofafrom the first row. M ij is often called theij th minor ofa. Theorem 6 The determinant of a matrix can be obtained by expansion from any row or column, where each term is the product of the element of the matrix, the corresponding minor, and either or - according to the chessboard pattern Example = ( 2) (expansion from first row) 4 = ( 2) ( 3) (expansion from 2 4 second column) 4

5 Example = ( ) ( 2) (expansion from second row) = ( 3) 4 2 (expansion from fourth column) 2 3 where We can also write the expansion of A from theith row as A =a i C i +a i2 C i2 +a i3 C i a in C in, C ij = ( ) i+j M ij. C ij is called theij th cofactor of the matrixa. The ( ) i+j automatically gives the correct sign + or. The expansion of A from thej th column is A =a j C j +a 2j C 2j +a 3j C 3j +...+a nj C nj. Definition 9 An upper triangular matrix is a square matrix in which every element below the the main diagonal is zero. A lower triangular matrix is a square matrix in which every element above the the main diagonal is zero. Theorem If a matrix is upper triangular or lower triangular then its determinant is the product of its diagonal elements. Proof. This can be proved by Induction using the expansion from the first row (in the case of lower triangular matrices) or the first column (in the case of upper triangular matrices). Corollary IfI is ann n identity matrix, then I =. 5

6 3 Properties of Determinants Theorem 2 For any square matrixa, A = A. i.e. The determinant of the transpose ofaequals the determinant ofa. Proof. This follows from the fact that the row expansion of a determinant is equal to a column expansion. The proof is by Induction. Theorem 3 If two rows (or columns) of a square matrixaare interchanged the determinant is multiplied by -. Proof. This can be proved by Induction. For 2 2 matrices c d a b =cb da = (ad bc) = a b c d. The Inductive step can be proved by expansion from any row that is fixed. Corollary 4 If two rows (or columns) of a square matrixaare identical, then A =. Proof. If we interchange the two rows that are identical, then the determinant is multiplied by -, by the Theorem. However, the matrix itself remains unchanged. Therefore, A = A 2 A = A =. Linearity Properties. 6

7 Theorem 5 If one row (or one column) of a square matrixais multiplied by a scalarλ, then the determinant is multiplied byλ. Proof. This follows from the row (or column) expansion. Corollary 6 If any row or column of a square matrix consists of only zeros then its determinant is zero. If one row (column) is a multiple of another row (column) then the determinant is zero. Proof. In class. Theorem 7 LetA,B andc ben n matrices such thatc ij =a ij +b ij for elements in the r th row (i.e. i =r, j =,2,...,n), and c ij =a ij =b ij for all other elements (i.e. i = r). Then C = A + B. Proof. This follows from the row expansion of the determinant from row r. Note that the above theorem also holds when the r th column of C is the sum of therth columns ofaandb and all other elements in the three matrices are equal. Example 8 a+b c+d e+f = a c e b d f Corollary 9 If a multiple of one row (column) of a square matrix is added to another row (column), the determinant remains unchanged Proof. We will prove this in the case where a multiple a rowr is added to the first row. All other cases and the corresponding result for columns are similar. a +λa r a 2 +λa r2... a n +λa rn a 2 a a 2n... a n a n2... a nn 7 = a a 2... a n a 2 a a 2n... a n a n2... a nn

8 λa r λa r2... λa rn a + 2 a a 2n... a n a n2... a nn = A + = A. This determinant is zero because one row is a multiple of another. 3. Simplifying and factorizing determinants Using row or column expansions to evaluate determinants is only feasible for determinants of small size, e.g. 2 2 or 3 3 determinants. For larger determinants a more practical method is to use elementary row or column operations to reduce the matrix to upper or lower triangular form. The effect of each operation is known, and since determinants of upper or lower triangular matrices can be easily computed, the determinant of the initial matrix can be expessed in terms of the determinant of the reduced matrix. Example R = R R = 3 = R R R = 3 = 3 3 ( 24) = Example 2 Vandermonde Determinants 8

9 a a 2 b b 2 c c 2 R = R = (b a)(c a) = (b a)(c a) a a 2 b a b 2 a 2 c a c 2 a 2 = a a 2 (b+a) (c+a) (b+a) (c+a) a a 2 b a (b a)(b+a) c a (c a)(c+a) expanding using the first column. = (b a)(c a){(c+a) (b+a)} = (b a)(c a)(c b) = (a b)(b c)(c a) in a more symmetric form. Example 22 a a b b 2 a c c 2 = a a a 2 a b b 2 = a a a a c c 2 a a 2 b b 2 c c 2 a a 2 = b b 2 the Vandermonde Determinant c c 2 = (a b)(b c)(c a). Example 23 a+b+c b+c 2a+b+2c a+b+2c R = a+b+c b+c a+c a+c = (a+c) a+b+c b+c = (a+c){(a+b+c) (b+c)} =a(a+c). 9

10 4 Determinants and Inverses Theorem 24 Forn n matricesaandb Proof. Omitted. AB = A B. Corollary 25 IfAhas an inverse then A =, and A = A = A. Proof. AA = I A A = Clearly, A = and A = A = A. Corollary 26 IfAis orthogonal then A =±. Definition 27 A matrix which has an inverse is a nonsingular matrix. A matrix which has no inverse is asingular matrix. Corollary 25 tells us that if a matrix is nonsingular then it has nonzero determinant. We would like to prove the converse of this. We recall that thei,j-th cofactor ofais C ij = ( ) i+j M ij where M ij is thei,j-th minor. Theorem 28 (Wrong Cofactor Expansion) IfAis ann n matrix A, ifr =i, a i C r +a i2 C r2 +a i3 C r a in C rn =, ifr=i.

11 Proof. In class. This result says that if we use the elements of one row together with the cofactors of a different row, the result is zero. Definition 29 The matrix of cofactors of A is the matrix whose i,j-th element isc ij, thei,j-th cofactor ofa. The adjoint of A, denoted by adja, is the transpose of the matrix of cofactors. Theorem 3 For any square matrixa, Proof. In class. A(adjA) = (adja)a = A I. Corollary 3 If A =, thenahas an inverse and A = A adja. Proof. A =, then from Theorem 3 This shows that the inverse ofais (adja)a = A I A adja A = I. A = A adja. Corollary 32 A square matrixais nonsingular if and only if A =. Proof. This follows from Corollary 25 and Corollary 3

12 5 Determinants and Systems of Linear Equations Consider the system Ax =b whereais a square matrix. IfAis non singular we can write A Ax = A b x = A b. The solution is unique in this case. Theorem 33 IfAis ann (n ) matrix, then the system Ax =b is consistent if and only if the augmented matrix A.b is singular. Proof. In class. We should note that computing the inverse of a square matrix by finding the adjoint and the determinant is not a good computational method. Gaussian elimination is far superior. Another bad computational method (except for small systems) for solving a system of equations is Cramer s Rule. Theorem 34 (Cramer s Rule) Consider the system Ax =b where A is a nonsingular square matrix. Let A = A.A 2.A 3..A n i.ea,a 2,A 3,,A n are the columns ofa. Then, x s = A.A 2.A 3. A s.b.a s+..a n. A 2

13 Proof. Let C s,c 2s,C 3s,...,C ns be the cofactors of the s th column of A. Then b C s +b 2 C 2s +b 3 C 3s +...+b n C ns = C s C 2s C 3s C ns b = Cs C 2s C 3s C ns Ax = A x =x s A. s th column. SinceAis nonsingular, A = and we have x s = b C s +b 2 C 2s +b 3 C 3s +...+b n C ns. A Clearly the numerator is the determinant of the matrix obtained by replacing columnsofaby the vector b. 3