Inverse matrix Definition: A has an inverse B if and only if AB=BA=I. THEOREM. An inverse is unique. THEOREM. If A has an inverse, then A is square. Non-square matrices don't have an inverse. THEOREM. The zero matrix does not have an inverse. THEOREM. Every elementary matrix E has an inverse. It is found as follows: Elementary Matrix Inverse Matrix combo(s,t,c) combo(s,t,-c) mult(t,m) mult(t,1/m) swap(s,t) swap(s,t)

How to compute the inverse matrixDef: AB=BA=I means B is the inverse of A. Inverse = adjugate/determinant (2x2 case). See the theorem below for 2x2. Inverse from the fundamental theorem on frame sequences. Frame sequences method. See the theorem below. Inverse rules Web References: Construction of inverses. Theorems on inverses.THEOREM. A square matrix A has a inverse if and only if one of the following holds: 1. rref(A) = I 2. Ax=0 has unique solution x=0. 3. det(A) is not zero. 4. rank(A) = n =row dimension of A. 5. There are no free variables in the last frame. 6. All variables in the last frame are lead variables. 7. nullity(A)=0.THEOREM. The inverse matrix is unique and written A^(-1).THEOREM. If A, B are square and AB = I, then BA = I.THEOREM. The inverse of inverse(A) is A itself.THEOREM. If C and D have inverses, then so does CD and inverse(CD) = inverse(D) inverse(C).THEOREM. The inverse of a 2x2 matrix is given by the formula 1 [ d -b] ------- [ ] ad - bc [-c a]THEOREM. The inverse B of any square matrix A can be found from the frame sequence method augment(A,I) toolkit steps combo, swap, mult . . augment(I,B) in which the inverse B of A is read-off from the right panel of the last frame.: Inverse matrix, frame sequence method (97.9 K, pdf, 03 Mar 2012)Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)Slides

Sample Exam: Exam 1 keys from S2010 and F2010.

More theorems on inversesTHEOREM. The inverse of inverse(A) is A itself.THEOREM. If C and D have inverses, then so does CD and inverse(CD) = inverse(D) inverse(C).

College Algebra Background:College algebra determinant definition Sarrus' rule for 2x2 and 3x3 matrices.

How to do 3.5-16 in maple. macro(combo=linalg[addrow]); A:=Matrix([[1,-3,-3],[-1,1,2],[2,-3,-3]]); B:=A^(-1); # expected answer Id:=Matrix([[1,0,0],[0,1,0],[0,0,1]]); A1:= < A | Id >; linalg[rref](A1); # Expected answer in right panel A2:=combo(A1,1,2,1); A3:=combo(A2,1,3,-2); See problem notes chapter 3

THEOREM. Homogeneous system with a unique solution. THEOREM. Homogeneous system with more variables than equations. Equation ideas can be used on a matrix A. View matrix A as the set of coefficients of a homogeneous linear system Ax=0. The augmented matrix B for this homogeneous system would be the given matrix with a column of zeros appended: B=aug(A,0). Answer checks matlab, maple and mathematica. Pitfalls.

General structure of linear systems.Superposition. General solution X=X_{0}+t_{1}X_{1}+ t_{2}X_{2}+ ... + t_{n}X_{n}. Question answered:What did I just do, by finding rref(A)?Problems 3.4-17 to 3.4-22 are homogeneous systems Ax=0 with A in reduced echelon form. Apply the last frame algorithm then write the general solution in vector form. EXAMPLE. A 3x3 matrix, frame sequence, Special Solutions of Gilbert Strang. How to find the vector general solution. How to find x_p: Set all free variable symbols to zero How to find x_h: Take all linear combinations of the Special Solutions. Superposition: x = x_p + x_h = General Solution

Methods for computing a determinantSarrus' rule, 2x2 and 3x3 cases. Four rules for determinantsTriangular Rule (one-arrow Sarrus' Rule): The determinant of a triangular matrix is the product of the diagonal elements.Multiply rule: B=answer after mult(t,m), then |A| = (1/m) |B|Swap rule: B=answer after swap(s,t), then |A| = (-1) |B|Combo rule: B=answer after combo(s,t,c), then |A| = |B|

Survey of Main theorems:Computation by the 4 rules, cofactor expansion, hybrid methods. Determinant product theorem det(AB)=det(A)det(B). Cramer's Rule for solving Ax=b:

x_{1}= delta_{1}/delta, ... , x_{n}= delta_{n}/delta Adjugate formula: A adj(A) = adj(A) A = det(A) I Adjugate inverse formula inverse(A) = adjugate(A)/det(A).Results on DeterminantsExamples: Computing det(A) easily. When does det(A)=0?THEOREM. Determinant values for elementary matrices: det(E)=1 for combo(s,t,c), det(E)=m for mult(t,m), det(E)=-1 for swap(s,t).

Discussion of 3.5 problems. Lecture Ideas of rank, nullity, dimension in examples.: Rank, nullity and elimination (154.1 K, pdf, 03 Mar 2012) More on Rank, Nullity dimension 3 possibilities elimination algorithmSlides

LectureCofactor expansion of det(A). minor(A,i,j) checkerboard sign (-1)^{i+j} cofactor(A,i,j)=(sign)minor(A,i,j) Details for 3x3 and 4x4. Hybrid methods to evaluate det(A). How to use the 4 rules to compute det(A) for any size matrix. Computing determinants of sizes 3x3, 4x4, 5x5 and higher. Frame sequences and determinants. Formula for det(A) in terms of swap and mult operations. Special theorems for det(A)=0 a zero row or col duplicates rows proportional rows. Elementary matrices Determinant product rule for an elementary matrix Cramer's rule. How to form the matrix of cofactors and its transpose. The adjugate matrix.

- det(triangular matrix)=the product of the diagonal elements, and
- det(EA)=det(E)det(A), where E is an elementary combo, swap or mult matrix.

Determinant product theoremdet(AB)=det(A)det(B) for any two square matrices A,B Proof details. Example.

Sample Exam: Exam 1 keys from S2010 and F2010.

Transpose matrix(A^T)^T = A (A + B)^T = A^T + B^T (AB)^T = B^T A^T

Further properties of the adjugate matrixComputing det(A) from A and adj(A) in 10 seconds Problems involving adj(A): examples from exams. Adjugate identity A adj(A) = adj(A) A = det(A) I 3x3 case: 6 ways to compute det(A) from A, adj(A). 3x3 case: the 6 cofactor expansions

Intro to Ch4Def: Vector==package of data items. Vectors are not arrows. The 8-Property Vector Toolkit Def: vector space, subspace Working set == subspace. Data set == Vector space Examples of vectors: Digital photos, Fourier coefficients, Taylor coefficients, Solutions to DE. Example: y=2exp(-x^2) for DE y'=-2xy, y(0)=2.