# 2250-1 7:30am Lecture Record Week 6 S2012

Last Modified: February 20, 2012, 21:36 MST.    Today: July 17, 2018, 17:23 MDT.

#### 13 Feb: Elementary matrices. Inverses. Sections 3.4, 3.5.

```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 matrix
Def: 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.
Slides: 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)```

#### Upcoming on 16 Feb:

Exam 1. Problems 1,2,3.
Sample Exam: Exam 1 keys from S2010 and F2010.
HTML: Exam links for the past 5 years (18.9 K, html, 02 May 2012)
```More theorems on inverses
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).
```
```College Algebra Background:
College algebra determinant definition
Sarrus' rule for 2x2 and 3x3 matrices.
```
References for 3.6 determinant theory and Cramer's Rule
Slides: Determinants 2008 (227.1 K, pdf, 03 Mar 2012)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)

## 14 Feb: Determinants, Survey of Results. Section 3.6

```  How to do 3.5-16 in maple.
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
```

html: Problem notes S2012 (5.2 K, html, 08 Apr 2012)
``` 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).
matlab, maple and mathematica.
Pitfalls.
```
``` General structure of linear systems.
Superposition.
General solution
X=X0+t1 X1 + t2 X2 + ... + tn Xn.
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 determinant
Sarrus' rule, 2x2 and 3x3 cases.
Four rules for determinants
Triangular 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|
```

## 15 Feb: Cramers Rule, Adjugate formula. Section 3.6

```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:
x1 = delta1/delta, ... , xn = deltan/delta
Adjugate formula: A adj(A) = adj(A) A = det(A) I
Results on Determinants
Examples: 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.
Slides: Rank, nullity and elimination (154.1 K, pdf, 03 Mar 2012)   More on Rank, Nullity
dimension
3 possibilities
elimination algorithm
```
```Lecture
Cofactor 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.
```
THEOREM. The 4 rules for computing any determinant can be compressed into two rules,
1. det(triangular matrix)=the product of the diagonal elements, and
2. det(EA)=det(E)det(A), where E is an elementary combo, swap or mult matrix.
```Determinant product theorem
det(AB)=det(A)det(B) for any two square matrices A,B
Proof details.
Example.
```

THEOREM. The adjugate formula A adj(A) = adj(A) A = det(A) I.
THEOREM. Adjugate inverse formula: inverse(A) = adj(A)/det(A).
Slides: Determinants 2010 (227.1 K, pdf, 03 Mar 2012)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (326.0 K, pdf, 07 Feb 2012)
html: Problem notes S2012 (5.2 K, html, 08 Apr 2012)

## 16 Feb: Exam 1

Exam 1. Problems 1,2,3.
Sample Exam: Exam 1 keys from S2010 and F2010.
HTML: Exam links for the past 5 years (18.9 K, html, 02 May 2012)

## 17 Feb: Problem session 3.4, 3.5, 3.6

```Transpose matrix
(A^T)^T = A
(A + B)^T = A^T + B^T
(AB)^T = B^T A^T
```
```Further properties of the adjugate matrix
Computing 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 Ch4
Def: 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.
```