Midterm I Review

Review for Midterm I

Math 20

October 16, 2007

Announcements
Midterm I 10/18, Hall A 7–8:30pm
ML Oﬃce Hours Wednesday 1–3 (SC 323)
SS review session Wednesday 8–9pm (location TBA)
Old exams and solutions on website

Outline

Determinants by sudoku
Vector and Matrix Algebra
patterns
Vectors
Determinants by Cofactors
Algebra of vectors
Systems of Linear Equations
Scalar Product
Matrices Inversion
Addition and scalar Determinants and the
multiplication of matrices inverse
Inverses of 2 × 2 matrices
Matrix Multiplication
The Transpose Computing inverses with the
Geometry adjoint matrix
Lines Computing inverses with row
Planes reduction
Determinants Properties of the inverse

Vector and Matrix Algebra
Learning Objectives

Add, scale, and compute linear combinations of vectors
Compute the scalar product of two vectors
Add and scale matrices
Multiply matrices
all of these with the caveat of “if possible.”

Vectors

Deﬁnition
An n-vector (or simply vector) is an ordered list of n numbers.
We can write them in a row or in a column.

1
b = (1, 0, −1)
2
a=
3

In linear algebra we mostly work with column vectors.

Algebra of vectors

Deﬁnition
Addition of vectors is deﬁned componentwise, as is scalar
multiplication:
   
a1 b1 a1 + b 1
 a2   b 2   a2 + b 2 
 . + . = . 
   
. .  . 
. . .
an bn an + bn
  
a1 ta1
a2  ta2 
t . = . 
  
.  . 
. .
an tan

Dot product

Deﬁnition
Given two vectors of the same dimension (size), their scalar
product (or “dot product”) is the sum of the product of the
components of the vectors:
  
a1 b1
n
a2  b2 
 .  ·  .  = a1 b1 + a2 b2 + · · · + an bn = ai bi
  
. .
. . i=1
an bn

Matrices

Deﬁnition
An m × n matrix is a rectangular array of mn numbers arranged in
m horizontal rows and n vertical columns.
 
a11 a12 · · · a1j · · · a1n
 a21 a22 · · · a2j · · · a2n 
 
. . . .
.. ..
. . . .
. .
. . . .
A=  ai1 ai2 · · · aij · · · ain 
 
. . . .
.. ..
. . . .
. .
. . . .
am1 am2 · · · amj · · · amn

Addition and scalar multiplication of matrices

Matrices can be added and scaled just like vectors:
Example

1 −1
12 21
+ =
34 02 36

Example

11 44
4 =
−1 2 −4 8

The matrix-vector product

Deﬁnition  
v1
v 
Let A = (aij ) be an m × n matrix and v =  2  a n-vector
. . . 
vn
(column vector). The matrix-vector product of A and v is the

w1
 w2 
vector Av =  , where
. . .
wm
n
wk = ak1 v1 + ak2 v2 + · · · + akn vn = akj vj ,
j=1

the dot product of kth row of A with v.

Example
Let
 
23
2
A = −1 4 and v=
−1
03

Find Av.

Example
Let
 
23
2
A = −1 4 and v=
−1
03

Find Av.

Solution
 
2 · 2 + 3 · (−1)
Av = (−1) · 2 + 4 · (−1)
0 · 2 + 3 · (−1)

Example
Let
 
23
2
A = −1 4 and v=
−1
03

Find Av.

Solution
  
2 · 2 + 3 · (−1) 4−1
Av = (−1) · 2 + 4 · (−1) = −2 − 4
0 · 2 + 3 · (−1) 0−3

Example
Let
 
23
2
A = −1 4 and v=
−1
03

Find Av.

Solution
  
2 · 2 + 3 · (−1) 4−1 1
(−1) · 2 + 4 · (−1) = −2 − 4 = −6 .
Av =
0 · 2 + 3 · (−1) 0−3 −3

Matrix product, deﬁned
Deﬁnition
Let A be an m × n matrix and B a n × p matrix. Then the matrix
product of A and B is the m × p matrix whose jth column is Abj .
In other words, the (i, j)th entry of AB is the dot product of ith
row of A and the jth column of B. In symbols
n
(AB)ij = aik bkj .
k=1

Example

   
1.5 0.5 1  125 115 110 105

 0 0.25 0 ‘70 60 50 40 5 7.5 7.5 7.5 
   
1.5 0.25 0  20 30 30 30 = 100 97.5 82.5 67.5
   
2 2 3 10 10 20 30 210 210 220 230 
3 2 2 270 260 250 240

The Transpose
There is another operation on matrices, which is just ﬂipping rows
and columns.
Deﬁnition
Let A = (aij )m×n be a matrix. The transpose of A is the matrix
A = (aij )n×m whose (i, j)th entry is aji .

The Transpose
and columns.
Deﬁnition
Example 
12
Let A = 3 4. Then
56

A=

The Transpose
and columns.
Deﬁnition
Example 
12
56

135
A= .
246

The Transpose
and columns.
Deﬁnition
Example 
12
56

135
A= .
246

Fact
Given matrices A and B, of suitable dimensions,

(AB) = B A

Geometry
Learning Objectives

Given a point and a line, decide if the point is on the line
Given a point and a direction, or two points, ﬁnd the equation
for the line
Given a point and a plane, decide of the point is on the plane
Given a point and a normal vector, ﬁnd the equation for the
plane

Lines

Deﬁnition
The line L through (the head of) a parallel to v is the set of all x
such that
x = a + tv
for some real number t.
The line L through (the heads of) a and b is

x = a + t(b − a)
= (1 − t)a + tb

Example
Find an equation for the line through (−2, 0, 4) parallel to
(2, 4, −2)

Example
Find an equation for the line through (−2, 0, 4) parallel to
(2, 4, −2)

Solution
x = (−2, 0, 4) + t(2, 4, −2)

Example
Find an equation for the line through (−3, 2, −3) and (1, −1, 4)

Example
Is (1, 2, 3) on this line?

Example
Is (1, 2, 3) on this line?
Answer.
No!

Planes

Deﬁnition
A hyperplane through (the head of) a that is orthogonal to a
vector p is the set of all (heads of) vectors x such that

p · (x − a) = 0

Example
Find an equation for the plane through (−3, 0, 7) perpendicular to
(5, 2, −1)

Example
(5, 2, −1)

Solution
5x + 2y − z = 22

Example
(5, 2, −1)

Solution
5x + 2y − z = 22

Question
Is (1, 2, 3) on this plane? Is there a point on the plane with z = 0?

Fact
If the equation for a plane is

Ax + By + Cz = D,

then p = (A, B, C ) is normal to the plane.

Determinants
Learning Objectives

Compute determinants of n × n matrices.

The determinant

Deﬁnition
a11 a12
The determinant of a 2 × 2 matrix A = is the number
a21 a22

a11 a12
= a11 a22 − a21 a12
a21 a22

Deﬁnition
The determinant of a 3 × 3 matrix is

a11 a12 a13
a21 a22 a23 = a11 a22 a33 − a11 a23 a32 − a21 a12 a33
a31 a32 a33
+ a21 a13 a32 + a31 a12 a23 − a31 a22 a13

Sarrus’s Rule

a11 a12 a13 a11 a22 a33 + a12 a23 a31
=
+a13 a21 a32 − a13 a22 a31
a21 a22 a23
−a11 a23 a32 − a12 a21 a33
a31 a32 a33

Sarrus’s Rule

a11 a12 a13 a11 a12 a11 a22 a33 + a12 a23 a31
=
+a13 a21 a32 − a13 a22 a31
a21 a22 a23 a21 a22
−a11 a23 a32 − a12 a21 a33
a31 a32 a33 a31 a32

Sarrus’s Rule

a11 a12 a13 a11 a12 a11 a22 a33 + a12 a23 a31
=
+a13 a21 a32 − a13 a22 a31
a21 a22 a23 a21 a22
−a11 a23 a32 − a12 a21 a33
a31 a32 a33 a31 a32

This trick does not work for any other determinants!

Determinants by sudoku patterns

Deﬁnition
Let A = (aij )n×n be a matrix. The determinant of A is a sum of
all products of n elements of the matrix, where each product takes
exactly one entry from each row and column.

Determinants by sudoku patterns

Deﬁnition
Let A = (aij )n×n be a matrix. The determinant of A is a sum of
all products of n elements of the matrix, where each product takes
exactly one entry from each row and column.
The sign of each product is given by (−1)σ , where σ is the number
of upwards lines used when all the entries in a pattern are
connected.

4 × 4 sudoku patterns

− − −
+ + +

− − −
+ + +

− − −
+ + +

− − −
+ + +

Determinants by Cofactors

Deﬁnition
Let A = (aij )n×n be a matrix. The (i, j)-minor of A is the matrix
obtained from A by deleting the ith row and j column. This matrix
has dimensions (n − 1) × (n − 1).
The (i, j) cofactor of A is the determinant of the (i, j) minor
times (−1)i+j .

The signs here seem complicated, but they’re not. The number
(−1)i+j is 1 if i and j are both even or both odd, and −1
otherwise. They make a checkerboard pattern:
 
+ − + ···
− + − · · ·
 
+ − + · · ·
 
. . . ..
... .
...

Fact
The determinant of A = (aij )n×n is the sum

a11 C11 + a12 C12 + · · · + a1n C1n

Fact

a11 C11 + a12 C12 + · · · + a1n C1n

Fact

a11 Ci1 + ai2 Ci2 + · · · + ain Cin

for any i.

Fact

a11 C11 + a12 C12 + · · · + a1n C1n

Fact

a11 Ci1 + ai2 Ci2 + · · · + ain Cin

for any i.

Fact

a1j C1j + a2j C2j + · · · + anj Cnj

for any j.

Theorem (Rules for Determinants)
Let A be an n × n matrix.
1. If a row or column of A is full of zeros, then |A| = 0.
2. |A | = |A|
3. If B is the matrix obtained by multiplying each entry of one
row or column of A by the same number α, then |B| = α |A|.
4. If two rows or columns of A are interchanged, then the
determinant changes its sign but keeps its absolute value.
5. If a row or column of A is duplicated, then |A| = 0.

Theorem (Rules for Determinants, continued)
Let A be an n × n matrix.
5. If a row or column of A is proportional to another, then
|A| = 0.
6. If a scalar multiple of one row (or column) of A is added to
another row (or column), then the determinant does not
change.
7. The determinant of the product of two matrices is the product
of the determinants of those matrices:

|AB| = |A| |B|

8. if α is any real number, then |αA| = αn |A|.

Systems of Linear Equations
Learning Objectives

Solve a System of Linear Equations using Gaussian Elimination
Find the (R)REF of a matrix.

Row Operations

The operations on systems of linear equations are reﬂected in the
augmented matrix.
1. Transposing (switching) rows in an augmented matrix does
not change the solution.
2. Scaling any row in an augmented matrix does not change the
solution.
3. Adding to any row in an augmented matrix any multiple of
any other row in the matrix does not change the solution.

Gaussian Elimination
1. Locate the first nonzero column. This is pivot column, and
the top row in this column is called a pivot position.
Transpose rows to make sure this position has a nonzero entry.
If you like, scale the row to make this position equal to one.
2. Use row operations to make all entries below the pivot
position zero.
3. Repeat Steps 1 and 2 on the submatrix below the first row
and to the right of the first column. Finally, you will arrive at
a matrix in row echelon form. (up to here is called the
forward pass)
4. Scale the bottom row to make the leading entry one.
5. Use row operations to make all entries above this entry zero.
6. Repeat Steps 4 and 5 on the submatrix formed above and to
the left of this entry. (These steps are called the backward
pass)

Example
Solve the system of linear equations

2x+ 8y +4z =2
2x+ 5y + z =5
4x+10y −4z =1

Inversion
Learning Objectives

Use the determinant to determine whether a matrix is
invertible
Find the inverse of a matrix

Determinants and the inverse

If A has an inverse A−1 , what is |A|−1 ?
Answer.

|A| A−1 = AA−1 = |I| = 1

Fact
A is invertible if and only if |A| = 0.

Inverses of 2 × 2 matrices

ab
Let A = . This is small enough that we can explicitly solve
cd
for A−1 .
Fact
If ad − bc = 0, then
−1
d −b
1
ab
= .
−c a
cd ad − bc

Cofactors

Remember how we computed determinants: Given A, Cij was
(−1)i+j times the determinant of Aij , which was A with the ith
row and j column deleted.
Let C+ = (Cij ). Then

A(C+ ) = |A| I

So
1
A−1 = (C+ )
|A|
We denote by adj A the matrix (C+ ) , the adjoint matrix of A.

Computing inverses with row reduction

To ﬁnd the inverse of A, form the augmented matrix A I and
row reduce. If the RREF has the form I B then B = A−1 .
Otherwise, A is not invertible.

Example  
0 2 −3
Find the inverse of A = −2 1 2  using row reduction.
201

0 2 −3 1 0 0 ←
−
  
20 1001
 −2 1  −2 1 2 0 1 0 ← +
−
2 0 1 0
0 2 −3 1 0 0
20 1001 ← −
 
20 1001
0 1 3 0 1 1  −2
0 2 −3 1 0 0 ← +−
 
20 10 0 1
0 1 30 1 1
0 0 −9 1 −2 −2 ×−1/9

1 ← − −+
 −−

201 0 0
1 ← +
−
0 1 3 0 1
0 0 1 −1/9 2/9 2/9
−3 −1
 ×1/2

−2/9
2 00 1/9 7/9
0 10 1/3 1/3 1/3 
0 1 −1/9
0 2/9 2/9
 
0 0 1/18 −2/18
1 7/18
0 10 1/3 1/3 1/3 
0 1 −1/9
0 2/9 2/9

Notice we get the same matrix as with the adjoint method.

Properties of the inverse

Theorem (Properties of the Inverse)
Let A and B be invertible n × n matrices. Then
(a) (A−1 )−1 = A
(b) (AB)−1 = B−1 A−1
(c) (A )−1 = (A−1 )
(d) If c is any nonzero number, (cA)−1 = c −1 A−1 .

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