# Where are we?

## Linear transformation

A\mathbf{x} = \mathbf{b}
A\mathbf{x} = \mathbf{0}
A\mathbf{x} = \lambda\mathbf{x}
matrix
vector
vector
matrix
vector
vector
matrix
vector
vector
scalar

## Linear equations

(why? we will see soon)
(why? we will see soon)

# What have we mainly focussed on?

rank < n,m

### rows with all 0s after GE

1~solution
\infty~solutions
0~or~1~solution
0~or~\infty~solutions
\begin{bmatrix} ~~~&~~~&~~~\\ ~~~&~~~&~~~\\ ~~~&~~~&~~~\\ \end{bmatrix}
rank=m=n
rank=m < n
\begin{bmatrix} ~~~&~~~&~~~&~~~&~~~\\ ~~~&~~~&~~~&~~~&~~~\\ ~~~&~~~&~~~&~~~&~~~\\ \end{bmatrix}
\underbrace{~~~~~~~~~~~~~~~~~~}
\underbrace{~~~~~~~~~~~~~~~}
\begin{bmatrix} ~~~&~~~&\\ ~~~&~~~&\\ ~~~&~~~&\\ ~~~&~~~&\\ ~~~&~~~&\\ ~~~&~~~& \end{bmatrix}

### No Free columns

rank=n < m
\begin{bmatrix} ~~~&~~~&~~~&~~~&~~~\\ ~~~&~~~&~~~&~~~&~~~\\ ~~~&~~~&~~~&~~~&~~~\\ \end{bmatrix}

### Pivot columns

\underbrace{~~~~~~~~~~~~~}
\underbrace{~~~~~~~~~~~~~~~~~~~~}

### Free columns

\begin{bmatrix} ~~~&~~~&\\ ~~~&~~~&\\ ~~~&~~~&\\ ~~~&~~~&\\ ~~~&~~~&\\ ~~~&~~~& \end{bmatrix}
\underbrace{~~~~~~}

### Pivot

\underbrace{~~~~~~~~~}

# Properties of determinants

### Property 2. row exchanges reverse the sign of the determinant

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ det(A) = ad - bc
det(I)=1
P = \begin{bmatrix} 0&1&0\\ 1&0&0\\ 0&0&1\\ \end{bmatrix}
det(P) = -1
P = \begin{bmatrix} 0&0&1\\ 1&0&0\\ 0&1&0\\ \end{bmatrix}
det(P) = -(-1) = 1
det(A_{permute}) = (-1)^k det(A)

# Properties of determinants

### Property 3a.

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ det(A) = ad - bc
\begin{vmatrix} ta&tb\\ kc&kd\\ \end{vmatrix}
= kt\begin{vmatrix} a&b\\ c&d\\ \end{vmatrix}

### Property 3b.

\begin{vmatrix} a+a'&b+b'\\ c&d\\ \end{vmatrix}
= \begin{vmatrix} a&b\\ c&d\\ \end{vmatrix}
+ \begin{vmatrix} a'&b'\\ c&d\\ \end{vmatrix}
\begin{vmatrix} ta&tb\\ c&d\\ \end{vmatrix}
= t\begin{vmatrix} a&b\\ c&d\\ \end{vmatrix}
det(2A)=
2^ndet(A)

### (With these two properties we have covered linear combinations of rows)

det(A+B) \neq det(A) + det(B)
(generally not equal)

# Properties of determinants

### Property 4.  If $$A$$ has 2 equal rows, $$det(A) = 0$$

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ det(A) = ad - bc

### Proof:

A = A_{exchange} \implies det(A) = det(A_{exchange})

### But from property 2

det(A) = -det(A_{exchange})
\therefore det(A) = det(A_{exchange}) = 0

# Properties of determinants

### Property 5.  Elementary row operations on a matrix (as in GE) do not change the determinant of the matrix

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ det(A) = ad - bc

### Informal Proof:

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}
A'=\begin{bmatrix} a & b\\ c - la & d - lb \end{bmatrix}
r2 = r2 - l*r1
det(A')=\begin{vmatrix} a & b\\ c - la & d - lb \end{vmatrix}
=\begin{vmatrix} a & b\\ c & d \end{vmatrix} -\begin{vmatrix} a & b\\ la & lb \end{vmatrix}
=\begin{vmatrix} a & b\\ c & d \end{vmatrix} -l\begin{vmatrix} a & b\\ a & b \end{vmatrix}
=det(A) - l * 0 = det(A)
property 3b
property 3a

### Implication: If $$A = LU$$  then $$det(A) = det(U)$$

property 4

# Properties of determinants

### Property 6.  If $$A$$ has a row of zeroes then $$det(A) = 0$$

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ det(A) = ad - bc

### Proof:

A=\begin{bmatrix} a & b&\dots\\ c & d&\dots\\ \dots&\dots&\dots \end{bmatrix}
det(A') = t\cdot det(A)
property 3b

### Implication: If after GE, $$U$$ has a row of 0s then $$det(A) = det(U) = 0$$

from properties 5 & 6
if~t=0
Let~A'=\begin{bmatrix} ta & tb&\dots\\ c & d&\dots\\ \dots&\dots&\dots \end{bmatrix}
A'=\begin{bmatrix} 0 & 0&\dots\\ c & d&\dots\\ \dots&\dots&\dots \end{bmatrix}
det(A') = 0\cdot det(A)
= 0

# Properties of determinants

### Property 7.  If $$U$$ is an upper triangular matrix

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ det(A) = ad - bc

### Proof:

U=\begin{bmatrix} a_{11} & * & * & \dots\\ 0 & a_{22} & * & \dots\\ 0 & 0 & a_{33} &\dots \\ \dots&\dots&\dots&\dots \\ 0 & 0 & 0 & a_{nn} \\ \end{bmatrix}
continue elimination to get row reduced form - this does not change the diagonal elements
R=\begin{bmatrix} a_{11} & 0 & 0 & \dots\\ 0 & a_{22} & 0 & \dots\\ 0 & 0 & a_{33} &\dots \\ \dots&\dots&\dots&\dots \\ 0 & 0 & 0 & a_{nn} \\ \end{bmatrix}
det(U) = det(R)
by property 5
det(U) = a_{11}\cdot a_{22}\cdot a_{33}\dots a_{nn}

# Properties of determinants

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ det(A) = ad - bc

### Implication: Once you do GE, you have an easy way of computing $$det(A)$$

R=\begin{bmatrix} a_{11} & 0 & 0 & \dots\\ 0 & a_{22} & 0 & \dots\\ 0 & 0 & a_{33} &\dots \\ \dots&\dots&\dots&\dots \\ 0 & 0 & 0 & a_{nn} \\ \end{bmatrix}
det(R) = a_{11}\cdot a_{22}\cdot a_{33}\dots a_{nn}\cdot det(I)
by property 3a
I=\begin{bmatrix} 1 & 0 & 0 & \dots\\ 0 & 1 & 0 & \dots\\ 0 & 0 & 1 &\dots \\ \dots&\dots&\dots&\dots \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}
= a_{11}\cdot a_{22}\cdot a_{33}\dots a_{nn}
from properties 5 & 7

### Property 7.  If $$U$$ is an upper triangular matrix

det(U) = a_{11}\cdot a_{22}\cdot a_{33}\dots a_{nn}

# Properties of determinants

### Property 8.  $$det(A) = 0$$ when $$A$$ is singular

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ det(A) = ad - bc

### Go from $$A$$ to $$U$$ using GE

by property 7
(determinant does not change - by property 5)

### If there is a 0 pivot (diagonal element) in $$U$$, $$det(U) = 0$$

(remember, A is square)

# Properties of determinants

### Property 9.  $$det(AB) = det(A) det(B)$$

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ det(A) = ad - bc

### HW4

A^{-1}A=I
\therefore det(A^{-1}A)=det(I)
\therefore det(A^{-1})det(A)=1
\therefore det(A^{-1})=\frac{1}{det(A)}
det(A^2)=det(AA)
=det(A)det(A)
=det(A)^2

# Properties of determinants

### Property 10.  $$det(A^\top) = det(A)$$

A=\begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ det(A) = ad - bc