The cofactor of element a. denoted A is defined as +Mi, where the sign is determined from i and j 4=(-1)"M4 The value of the n X n determinant equals the sum of products of elements of any row(or column) and their respective cofactors. Thus, for the 3 x 3 determinant det A=a, Au+a,2 A+a,3A3(first row) =a,A,+a,, A,,+a,A(first column) 3. Properties of Determinants a. If the corresponding columns and rows of A are interchanged, det A is unchanged. b. If any two rows (or columns)are interchanged, the gn of det A changes. c. If any two rows(or columns)are identical, det A=0. d. If A is triangular (all elements above the main
17 The cofactor of element aij, denoted Aij, is defined as ±Mij, where the sign is determined from i and j: A M ij i j ij = − + ( )1 . The value of the n × n determinant equals the sum of products of elements of any row (or column) and their respective cofactors. Thus, for the 3 × 3 determinant, det A a A a A a A = + + (first row) 11 11 12 12 13 13 or = + + a A a A a A 11 11 21 21 31 31 (first column) etc. 3. Properties of Determinants a. If the corresponding columns and rows of A are interchanged, det A is unchanged. b. If any two rows (or columns) are interchanged, the sign of det A changes. c. If any two rows (or columns) are identical, det A = 0. d. If A is triangular (all elements above the main diagonal equal to zero), A = a11 · a22 ·…· ann: a a a a a a a n n n nn 11 21 22 1 2 3 0 0 0 0 0 … … … … … … … …
e. If to each element of a row or column there is added C times the corresponding element another row (or column). the value of the determi- 4. Matrice Definition. A matrix is a rectangular array of numbers and is represented by a symbol A or [a;: The numbers a; are termed elements of the matrix; ubscripts i and j identify the element, as the number row i and column j. The order of the matrix is mxn C"m by n). When m =n, the matrix is square and is aid to be of order n. For a square matrix of order n the elements au, az, .. an constitute the main diagonal. 5. Operations Addition: Matrices A and B of the same order may be added by adding corresponding elements, ie,A+B={a+bp小 Scalar multiplication: If A=[a l and c is a constant (scalar), then cA =[cai l, that is, every element of A is multiplied by c. In particular, (-1)A=-A Fa;l and A+(A)=0, a matrix with all elements ual to zero
18 e. If to each element of a row or column there is added C times the corresponding element in another row (or column), the value of the determinant is unchanged. 4. Matrices Definition. A matrix is a rectangular array of numbers and is represented by a symbol A or [aij]: A a a a a a a a a a n n m m m n = … … … … … … … 11 12 1 21 22 2 1 2 = [ ] aij The numbers aij are termed elements of the matrix; subscripts i and j identify the element, as the number is row i and column j. The order of the matrix is m × n (“m by n”). When m = n, the matrix is square and is said to be of order n. For a square matrix of order n the elements a11, a22, …, ann constitute the main diagonal. 5. Operations Addition: Matrices A and B of the same order may be added by adding corresponding elements, i.e., A + B = [(aij + bij)]. Scalar multiplication: If A = [aij] and c is a constant (scalar), then cA = [caij], that is, every element of A is multiplied by c. In particular, (–1)A = –A = [–aij] and A + (–A) = 0, a matrix with all elements equal to zero
Multiplication of matrices: Matrices A and B may be multiplied only when they are conform- able, which means that the number of columns of A equals the number of rows of B. Thus, if A is m x k and B is k x n, then the produc C=AB exists as an m x n matrix with elements qual to the sum of products of elements in row i of A and corresponding elements of column j of B: For example, if then element cu is the sum of products ambu+ a2b21+…+a2b 6. Properties A+B=B+A A+(B+C)=(A
19 Multiplication of matrices: Matrices A and B may be multiplied only when they are conformable, which means that the number of columns of A equals the number of rows of B. Thus, if A is m × k and B is k × n, then the product C = AB exists as an m × n matrix with elements cij equal to the sum of products of elements in row i of A and corresponding elements of column j of B: c a b ij il lj l k = = ∑1 For example, if a a a a a a a a k k m m k 11 12 1 21 22 2 1 … … … … … … … … ⋅ … … … … … … … b b b b b b b b b n n k k k n 11 12 1 21 22 2 1 2 = … … … … … … c c c c c c c n n m 11 12 1 21 22 2 1 c c m2 … m n then element c21 is the sum of products a21b11 + a22b21 + … + a2kbk1. 6. Properties A B B A A B C A B C + = + + + = + + ( ) ( )
(c1+c,)A C(A+B)=cA+cB (c2A)=(c1c2)A (AB)(C)=A(BC) (A+B)(C)=AC + BC AB≠BA( in general) 7. Transpose If a is an n x m matrix. the matrix of order m x n obtained by interchanging the rows and columns of A is called the transpose and is denoted A'. The following are properties of A, B, and their respective transposes (A) (A+B)=A+B (AB)=BA A symmetric matrix is a square matrix a with the 8. Identiny Matrix A square matrix in which each element of the main agonal is the same constant a and all other elements ro is called a scalar matrix
20 ( ) ( ) ( ) ( ) c c A c A c A c A B cA cB c c A c c A 1 2 1 2 1 2 1 2 + = + + = + = (AB C A BC A B C AC BC AB BA ) ) ( ) ( ) ( ) in general ( ( = + = + ≠ ) 7. Transpose If A is an n × m matrix, the matrix of order m × n obtained by interchanging the rows and columns of A is called the transpose and is denoted AT. The following are properties of A, B, and their respective transposes: ( ) ) A A A B A B cA cA AB B A T T T T T T T T T T = + = + = = ( ( ) ( ) A symmetric matrix is a square matrix A with the property A = AT. 8. Identity Matrix A square matrix in which each element of the main diagonal is the same constant a and all other elements zero is called a scalar matrix
000 When a scalar matrix multiplies a conformable second matrix A, the product is aA, that is, the same as multi- plying A by a scalar a. A scalar matrix with diagonal is denoted 1. Thus, for any nth-order matrix A, the identity matrix of order n has the property 9. Adjoint If A is an n-order square matrix and Ai the cofactor of element ij, the transpose of (A l is called the adjoint 10. Inverse Matrix Given a square matrix A of order n, if there atrix B such that ab= BA= then B is cal Iverse of A. The inverse is denoted A-LA sary and sufficient condition that the square matrix A have an inverse is det a+0. Such a matrix is called nonsingular; its inverse is unique and is given by
21 a a a a 0 0 0 0 0 0 0 0 0 0 0 0 0 … … … … … … … … When a scalar matrix multiplies a conformable second matrix A, the product is aA, that is, the same as multiplying A by a scalar a. A scalar matrix with diagonal elements 1 is called the identity, or unit matrix, and is denoted I. Thus, for any nth-order matrix A, the identity matrix of order n has the property AI IA A = = 9. Adjoint If A is an n-order square matrix and Aij the cofactor of element aij, the transpose of [Aij] is called the adjoint of A: adj A Aij T = [ ] 10. Inverse Matrix Given a square matrix A of order n, if there exists a matrix B such that AB = BA = I, then B is called the inverse of A. The inverse is denoted A–1. A necessary and sufficient condition that the square matrix A have an inverse is det A ≠ 0. Such a matrix is called nonsingular; its inverse is unique and is given by