Edwards-Penney Chapter 3, sections 3.4, 3.5, 3.6 Topics, Definitions, Theorems 3.4 ==== Def: Matrix add, Matrix scalar multiply Def: Vector, Matrix Multiply Def: Equality of matrices Algorithm. Changing Ax=b to equations and equations to Ax=b. Theorem 1. Matrix algebra A+B=B+A A+(B+C)=(A+B)+C A(BC)=(AB)C A(B+C)=AB+AC, (D+E)F=DF+EF Def: Zero matrix 0 Theorems: 0+A=A+0, A0=0, 0A=0 Def: Identity matrix I Theorems: AB=AC ==> B=C is false AI=A, IB=B Def: Inverse matrix of A is a matrix B such that AB=BA=I. 3.5 ==== Def: Augmented matrix of vectors [v1 v2 v3 ... vn] Same as Maple v:= for n=3 Theorems: Ax in terms of columns of A AB in terms of columns of B Def: Matrix A has inverse matrix B means AB=BA=I Theorem 1. The inverse of a matrix is unique. When it exists, then write B = A^(-1) Theorem 2. Inverse of a 2x2 matrix A:=Matrix([[a,b],[c,d]]) equals B=(1/det(A))Matrix([[d,-b],[-c,a]]) Theorem 3. (a) (A^(-1))^(-1)=A (b) (A^n)^(-1)=(A^(-1))^n for integer n>=0 (c) (AB)^(-1)=B^(-1) A^(-1) Theorem 4. The nxn system Ax=b with A invertible has unique solution x=A^(-1)b Def: Elementary matrix E = answer after one toolkit operation of combo, swap or mult applied to the identity matrix I. Theorem 5. Let one toolkit operation be applied to matrix A to obtain matrix B. Let E be the elementary matrix obtained from the identity by applying the identical toolkit operation. Then EA=B. Theorem 6. An nxn matrix A is invertible if and only if there is a sequence of elementary matrices E_1 to E_k such that the product E_k ... E_1 A = I. ALGORITHM. To find the inverse of A, find toolkit operations to transform A to the identity. Then do the same operations, same order, to I. The result will be A^(-1). METHOD. Form C=. Find the last frame . Then B is the inverse of A. Theorem. To solve AX=B where X is a matrix with many columns, multiply by the inverse of A to get X=A^(-1)B. Find X from the augmented matrix C= by toolkit operations to . Theorem 7. The following peoperties of a square nxn matrix A are equivalent. (a) A has an inverse (b) A transforms by toolkit operations to the identity I. (c) Ax=0 has only the sollution x=0. (d) Ax=b has a unique solution x for each possible b. (e) Ax=b has a solution x for every possible b. (f) det(A) is nonzero [See section 3.6] 3.6 ==== Def: Sarrus' rule for 2x2 determinants Def: Value of a 3x3 determinant Equivalent to Sarrus' 3x3 rule with 6 arrows. Def: Minor and cofactor. Def: det(A) for nxn matrix A, defined recursively. Theorem 1. Cofactor expansion by rows or columns. Theorem. (a) If mult(t,m) is applied to matrix A to obtain matrix B, then det(B)=m det(A). (b) If swap(s,t) is applied to matrix A to obtain matrix B, then det(B) = (-1) det(A). (c) If combo(s,t,c) is applied to matrix A to obtain matrix B, then det(B)=det(A). (d) If matrix A has two row identical (or two columns), then det(A)=0. (e) If matrices A1, A2, B are identical except for one row i and row(A1,i)+row(A2,i)=row(B,i), then det(A1)+det(A2)=det(B). Def: upper triangular, lower triangular, triangular Theorem: det(triangular matrix) = product of the diagonal elements Def: Transpose of a matrix A^T Theorem: (i) (A^T)^T = A (ii) (A+B)^T = A^T + B^T (iii) (cA)^T = c A^T (iv) (AB)^T = B^T A^T Theorem 2. Given a square matrix A. Then A^(-1) exists if and only if det(A) is nonzero. Theorem 3. Let A,B be square matrices. Then det(AB)=det(A)det(B). Theorem 4. Cramer's rule for solving Ax=b. Theorem 5. A^(-1) = adjoint(A)/det(A)