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Section Two: Matrices

Textbook: Ch. 1.8, 1.10GOALS OF THIS CHAPTER

- develop notion of matrix inverse

- discuss properties of matrix inverse

- see elementary row operation procedure of finding an inverse

- introduce cofactors and the adjoint of a matrix

- see the adjoint formula for finding an inverse

- understand what the existence of an inverse tells usabout linear systems

- understand and calculate elementary matrices

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WHAT IS A MATRIX INVERSE?

When dealing with numbers, we have anunderstanding of what the inverse of a number

a is: it is the unique number b that satisfiesthe inverse equation

We say b is the inverse of a (provided it exists). We can evensolve for b:

ab = 1

b = 1/a

So we can write the inverse equation as:

a * 1/a = aa-1= 1

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WHAT IS A MATRIX INVERSE?

It is natural to wonder if there is such a thingas an inverse for a matrix A. The answer is

maybe. If there is an inverse (call it B), it mustsatisfy the matrix inverse equations:

Usually, we write A-1 instead of B, so that the matrix inverseequations look like:

AB = I

If the inverse does not exist, we say A is singular(or non-invertible). If the inverse does exist, thenwe say A is non-singular (or invertible).

BA = I

AA-1 = I A-1A = I

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WHAT IS A MATRIX INVERSE?

Thm. 1 Uniqueness of Inverse

- done in classProof - done in class

Thm. 2 Properties of Matrix Inverse- done in class

Proof - done in class

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ELEMENTARY ROW OPERATION PROCEDURE

Next, I will guide you through the steps needed to find

the inverse of a square matrix A. This method willeither find the inverse, or it will tell you that theinverse does not exist.

A =

Ex. 3 Finding the Inverse using E.R.O.

1 2 3

1 1 2

0 1 2

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ELEMENTARY ROW OPERATION PROCEDURE

1 2 3

1 1 2

0 1 2

STEP #1: Adjoin an identity

matrix of the same size to the

right hand side of A. We callthis matrix [A|I].

1 0 0

0 1 0

0 0 1

STEP #2: Use elementary row

operations to reduce the left-

hand side of [A|I] to RREF.

Ex. 3 Finding the Inverse using E.R.O.

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ELEMENTARY ROW OPERATION PROCEDURE

1 2 3

1 1 2

0 1 2

1 0 0

0 1 0

0 0 1

R2 R2 R1

1 2 3

0 -1 -1

0 1 2

1 0 0

-1 1 0

0 0 1

R2 (-1)R2

Ex. 3 Finding the Inverse using E.R.O.

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ELEMENTARY ROW OPERATION PROCEDURE

1 2 3

0 1 1

0 1 2

1 0 0

1 -1 0

0 0 1 R3 R3 R2

1 2 3

0 1 1

0 0 1

1 0 0

1 -1 0

-1 1 1

R1 R1 2R2

Ex. 3 Finding the Inverse using E.R.O.

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ELEMENTARY ROW OPERATION PROCEDURE

1 0 1

0 1 1

0 0 1

-1 2 0

1 -1 0

-1 1 1

R1 R1 R3

1 0 0

0 1 1

0 0 1

0 1 -1

1 -1 0

-1 1 1

R2 R2 R3

Ex. 3 Finding the Inverse using E.R.O.

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ELEMENTARY ROW OPERATION PROCEDURE

1 0 0

0 1 0

0 0 1

0 1 -1

2 -2 -1

-1 1 1

RREF

STEP #3: If the RREF of A is an identity matrix, then A has an inverse

and the inverse is the matrix on the right hand side!

A-1

[A|I] E.R.O. [I|A-1]

Ex. 3 Finding the Inverse using E.R.O.

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ELEMENTARY ROW OPERATION PROCEDURE

1 0 1

0 1 0

0 0 0

0 1 -1

2 -2 -1

-1 1 1

RREF (full row

of zeros)

A-1 does not exist

(even though there is

a matrix over here)

Ex. 3 Finding the Inverse using E.R.O.

The only other thing that may happen, is you obtain a full row ofzeros along the way. If this happens, then A is not invertible (orA is singular).

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MINORS AND COFACTORS

For a 2x2 matrix, we can calculate a special number

called the determinant. We denote it as det(A).

A =

Ex. 4 The 2x2 Determinant

1 2

3 4

Multiply entries on main diagonal.

Subtract the product of the

remaining entries.

det(A) = 1*4 2*3

= -2

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1 2 23 0 5

0 -7 2

MINORS AND COFACTORS

Let A be a square matrix. The minor matrix of theentry a(i,j) is the new matrix formed by deleting the

ith row and jth column of A. The minor of a(i,j) is thedeterminant of the minor matrix, denoted as Mij.

A =

Ex. 5 Minors of a 3x3 Matrix

M11 =0 5

-7 2

The minor of the (1,1) entry.

det = 0 (-35) = 35

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1 2 23 0 5

0 -7 2

MINORS AND COFACTORS

A =

Ex. 5 Minors of a 3x3 Matrix

M23 =1 2

0 -7

The minor of the (2,3) entry.

= -7 0 = -7det

Let A be a square matrix. The minor matrix of theentry a(i,j) is the new matrix formed by deleting the

ith row and jth column of A. The minor of a(i,j) is thedeterminant of the minor matrix, denoted as Mij.

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1 2 23 0 5

0 -7 2

MINORS AND COFACTORS

A =

Ex. 5 Minors of a 3x3 Matrix

M32 =1 2

3 5

The minor of the (3,2) entry.

= 5 6 = -1det

Let A be a square matrix. The minor matrix of theentry a(i,j) is the new matrix formed by deleting the

ith row and jth column of A. The minor of a(i,j) is thedeterminant of the minor matrix, denoted as Mij.

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MINORS AND COFACTORS

For any square matrix, we denote the cofactor of an

entry as Aij. The cofactor is defined to be

Aij = (-1)i+jMij

1 2 2

3 0 5

0 -7 2

A =

Ex. 6 Cofactors of a 3x3 Matrix

A23 = (-1)2+3M23

M23 =1 2

0 -7

= (-1)5(-7-0)

= (-1)(-7)

= 7

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MINORS AND COFACTORS

For any square matrix, we denote the cofactor of an

entry as Aij. The cofactor is defined to be

Aij = (-1)i+jMij

1 2 2

3 0 5

0 -7 2

A =

Ex. 6 Cofactors of a 3x3 Matrix

A11 = (-1)1+1M11

= (-1)2(0+35)

= 35

M11 =0 5

-7 2

The cofactor returns a

SINGLE NUMBER.

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THE ADJOINT MATRIX

There is one cofactor for each entry in an nxn square

matrix A. If we combine all these cofactors, we obtainthe cofactor matrix, or cofmat(A).

cofmat(A) =

A11 A12 A1n

A21 A22 A2n

An1 An2 Ann

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(-1)2(4)B11

B12

B21 B22

THE ADJOINT MATRIX

cofmat(B) =

Ex. 7 2x2 Cofactor Matrix

B =

1 2

3 4

=

Since there isnt a matrix left over

when we cross out the column and

row, we just write down the

number left over.

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(-1)2(4) (-1)3(3)B11

B12

B21 B22

THE ADJOINT MATRIX

cofmat(B) =

Ex. 7 2x2 Cofactor Matrix

B =

1 2

3 4

=

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(-1)2(4) (-1)3(3)

(-1)3(2)

B11

B12

B21 B22

THE ADJOINT MATRIX

cofmat(B) =

Ex. 7 2x2 Cofactor Matrix

B =

1 2

3 4

=

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(-1)2(4) (-1)3(3)

(-1)3(2) (-1)4(1)

B11

B12

B21 B22

THE ADJOINT MATRIX

cofmat(B) =

Ex. 7 2x2 Cofactor Matrix

B =

1 2

3 4

=

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4 -3

-2 1

B11

B12

B21 B22

THE ADJOINT MATRIX

cofmat(B) =

Ex. 7 2x2 Cofactor Matrix

B =

1 2

3 4

=

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THE ADJOINT MATRIX

Ex. 8 3x3 Cofactor Matrix

- done in class

Once we know the matrix of cofactors, all we have to do is take thetranspose of the matrix of cofactors to get to the adjoint.

adj(A) = cofmat(A)T

Ex. 9 3x3 Adjoint Matrix (using Ex. 8)- done in class

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THE ADJOINT MATRIX

Why does the adjoint matter? Well, it turns

out we can find the inverse of an nxn matrix Ausing the adjoint formula.

A-1 = 1/det(A) adj(A)

So, if we know the adjoint of a square matrix A, wejust scalar multiply by one over the determinant toget the inverse. Right now, we can only use this for2x2 matrices (we will learn how to finddeterminants of bigger matrices later).

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THE ADJOINT MATRIX

This formula gives us a shortcut to find the

inverse for a 2x2 matrix.

A-1 =

A =

a b

c d

d -b

-c a

1

ad-bc

Adjoint of AOne over the

determinant

Just scalar multiply

these two pieces to get

the inverse of A!

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USING THE INVERSE TO SOLVE Ax=b

Suppose that the industrial process of a company can be modeled bythe matrix equation Ax=b. The input vector x represents the amountsof materials that are entered into the industrial process and the

output vector b represents the amounts of materials obtained afterthe process is complete. If the company would like the output of threematerials to be 15, 10 and 5; how much of the materials are requiredas input into the process?

Ex. 10 Solving a matrix system with the inverse

1 2 3

1 1 2

0 1 2

A =

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USING THE INVERSE TO SOLVE Ax=b

Ex. 10 Solving a matrix system with the inverse

0 1 -1

2 -2 -1

-1 1 1

A-1 =

If we know A is invertible, we can easily solve the matrix systemAx=b.

In Ex. 3, we found the inverse of A:

A-1Ax = A-1b

Ix = A-1

bx = A-1b

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USING THE INVERSE TO SOLVE Ax=b

Ex. 10 Solving a matrix system with the inverse

We can use information from the question to write out the outputvector b.

15

10

5

b =

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USING THE INVERSE TO SOLVE Ax=b

Ex. 10 Solving a matrix system with the inverse

The final answer is given by the multiplication x = A-1b:

xy

z

0 1 -12 -2 -1

-1 1 1

1510

5

=

=

5

5

0

So the company would

require 5 units of the first, 5

units of the second, and 0

units of the third input

materials.

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USING THE INVERSE TO SOLVE Ax=b

If we know that an nxn matrix A is non-singular, then

we also know many other things! We often call thefollowing list A List of Non-Singular Equivalences. Wewill add more to this list in future chapters.

(1) The nxn matrix A is invertible.(2) The system Ax=b has a unique solution for everyb

(3) The only solution of the homogeneous systemAx=0 is the trivial solution x=0

(4) A is row equivalent to I (ie. there is not a full

row of zeros in the RREF of A).

Proof to be completed in a future chapter

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ELEMENTARY MATRICES

A square matrix E is called an elementary matrix if it

can be obtained from the identity matrix by a singleelementary row operation.

1 0 0

0 1 00 0 1

1 0 0

0 1 00 0 2

TYPE I multiplying a row by a non-zero number

R3 2 R3

E1

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ELEMENTARY MATRICES

A square matrix E is called an elementary matrix if it

can be obtained from the identity matrix by a singleelementary row operation.

1 0 0

0 1 00 0 1

1 0 -3

0 1 00 0 1

TYPE II adding a multiple of one row to another

R1 R1 3R3

E2

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ELEMENTARY MATRICES

A square matrix E is called an elementary matrix if it

can be obtained from the identity matrix by a singleelementary row operation.

1 0 0

0 1 00 0 1

R2 R3

1 0 0

0 0 10 1 0

TYPE III switching two rows (called a permutation)

E3

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ELEMENTARY MATRICES

Ex. 11 Invertible Matrix as Product of Elementary Matrices

- done in class

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ELEMENTARY MATRICES

Thm. 12 Every elementary matrix is invertible. Theinverse of an elementary matrix is an elementarymatrix of the same type.

Proof not required

Thm. 13 A square matrix A is invertible if and only if it is theproduct of elementary matrices.

Proof done in class

Note: we can generalize Thm. 11 so that any squarematrix A can be written as a product of elementarymatrices and an upper triangular matrix. (seetextbook pg. 133) This is important since A need notbe invertible to do so!

Appendix B: If and Only If Proofs done in class