ENGR-1100 Introduction to Engineering Analysis

Lecture 6

Lecture outline

•Matrix and Matrix operations•Rules of matrix arithmetic•Determinants•2 In class problems

Definition: A matrix is a rectangular array of numbers

2 2

-2 0

-1 -1

2 2 π

-2 0 sin(2)

-1 -1 e

2 2 π −3

The size of the matrix m x n. Where m the number of rows and n the number of columns.

Capital letters denote matrices

Lower case letters denote numerical quantities

2 2

-2 0

-1 -1

A= 2 2 π −3C=

In a square matrix m = n a11 a12….… a1n

a21 a22….… a2n

: : :

an1 an2….… ann

A= a11, a22….… ann are on the main diagonal

Definition 1: equal matrices

• Matrix are said to be equal if they have: (i) the same size, and (ii) the corresponding entries in the two matrices are equal.

2 2

-2 0

-1 -1

A=

1 2

-2 0

3 -1

B=

2 2 1

-2 0 2

-1 -1 2

C=

A=B; B=C; A=C

Definition 2: sum of matrices• If A and B are any two matrices of the same size, then the

sum A+B is the matrix obtained by adding together the corresponding entries in the two matrices. Matrices of different sizes can’t be added.

2 2

-2 0

-1 -1

A=

1 2

-2 0

3 -1

B=

2 2 1

-2 0 2

-1 -1 2

C=

3 4

-4 0

2 -2

A+B=A+C and B+C are undefined

Definition 3: matrix scalar product• If A is any matrix and c any scalar, then the product cA is

the matrix obtained by multiplying each entry of A by c.

2 2

-2 0

-1 -1

A=

4 4

-4 0

-2 -2

2A=

-2 -2

2 0

1 1

(-1)A=-A=

Definition 4: Product of two matrices

1 2 4

2 6 0A=

4 1 4 3

0 -1 3 1

2 7 5 2

B=

A x B =?

=1 2 42 6 0

4

0

2

1

-1

7

4

3

5

3

1

2

26

(2x4)+(6x3)+(0x5)=26

=1 2 42 6 0

4

0

2

1

-1

7

4

3

5

3

1

2

13

26

(1x3)+(2x1)+(4x2)=13

Example 1 complete the product computation

=1 2 42 6 0

4

0

2

1

-1

7

4

3

5

3

1

2

13

26

Example 1 Solution

=1 2 42 6 0

4

0

2

1

-1

7

4

3

5

3

1

2

12 27 30 13

8 -4 26 12

120662x01x63x2c 3020645x43x24x1c40622x01x61x2c 2728217x41x21x1c

80082x00x64x2c ,128042x40x24x1c

2,43,1

2,22,1

2,11,1

=−+=++==++=++=

−=+−=+−+==+−=+−+=

=++=++==++=++=

The size of a product matrix

A B = AB

m x r r x n m x n

inside

outside

Remember, Rows then Columns. (Thankfully, your calculator and Matlab work like this too).

Transpose of a matrix AT

a11

a12

a13

a14

AT=

a21

a22

a23

a24

a31

a32

a33

a34

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

A=

Example 2 Determine ED

E=

1 5 2

-1 0 1

3 2 4

D=

6 1 3

-1 1 2

4 1 3

2112184x31x12x43,3f 266202x30x15x43,2f 129143x31x11x41,3f

78124x21x12x12,3f 1-4-52x21x0-1x52,2f 46113x21x11x11,2f

25121124x31x12x61,3f 366302x30x15x61,2f 149163x31x11x61,1f,EDFLet

=++=++==+=++==+−=+−+=

=++−=++−==+=++==+−−=+−+−=

=++=++==+=++==+−=+−+==

−=

212612714253614

ED

Example 3 Determine D+E

E=

1 5 2

-1 0 1

3 2 4

D=

6 1 3

-1 1 2

4 1 3

−=+

737312567

ED

Rules of matrix arithmetic

-1 0

2 3A=

1 2

3 0B=

Multiply AB:

-1 -2

11 4AB=

3 6

-3 0BA=

Then: AB=BA

Multiply BA

The following rules of matrix arithmetic are valid(assuming that the sizes of the matrices are such that the indicated operations can be performed)

• (a) A+B=B+A• (b) A+(B+C)=(A+B)+C• (c) A(BC)=(AB)C• (d) A(B±C)=AB±AC• (e) (B±C)A=BA±CA• (f) a(B±C)=aB±aC• (g) (a±b)C=aC±bC• (h (ab)C=a(bC)• (i) a(BC)=(aB)C=B(aC)

Identity matrix:square matrix with 1’s on the main diagonal and 0’s off the main diagonal

1 0

0 1I2=

1 0 00 1 00 0 1

I3=

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

I4=

If A is an mxn matrix, then:

AIn=A and ImA=A

Why study determinants?• They have important applications to system of

linear equations and can be used to produce formula for the inverse of an invertible matrix.

• The determinant of a square matrix A is denoted by det(A) or |A|. If A is 1x1 matrix

A=[a11]Then: det(A)= a11

For example A=[-7]det (A)=det(-7)=-7

Determinant of a 2x2 matrix

If A is a 2x2 matrixa11 a12

a21 a22A=

Then we define

a11 a12

a21 a22det(A)= = det(A)= a11a22 -a21a12

a11 a12

a21 a22

minus

Example 1

5 4

3 2A=

Then we define

5 4

3 2det(A)= = det(A)= (5x2)-(3x4)=-2

5 4

3 2

Duplicate column method – for 3x3

a11 a12 a13

a21 a22 a23

a31 a32 a33

a11 a12

a21 a22

a31 a32

det(A)=

1 5 -3

1 0 2

3 -1 2

Example1 5

1 0

3 -1

=0+30+3-0-(-2)-10=25

I personally do not like this method. It only works for 3 x3It means nothing physically.

Class assignment 1

-1 2 3

4 1 -6

-3 5 2

A=

• Find det(A) using the method shown in class

• Let

Class assignment 1 Solution

-1 2 3

4 1 -6

-3 5 2

A=

57)692032(]23*3[]10*2[)]32(*1[)]1*3()5*4[(*)3[(

)]6*3()2*4[(*)2[()]]6*5()2*1[(*)1[(

253614

321

=++−=+−−−=−−+

−−−−−−−=

−−

−=A

Inverse of a 2x2 matrix

If A is a 2x2 matrix

Then we define

𝐴𝐴 = 𝑎𝑎 𝑏𝑏𝑐𝑐 𝑑𝑑

𝐼𝐼𝐼𝐼𝐼𝐼 𝐴𝐴 = 𝐴𝐴−1 =𝑑𝑑 −𝑏𝑏−𝑐𝑐 𝑎𝑎det(𝐴𝐴)

So, if the determinant is zero, what does that say about A-1 ?

Example 2

5 4

3 2A=

Then we define

5 4

3 2det(A)= = det(A)= 5X2-4X3=-2

2 -4

-3 5Inv(A)= / -2

-1 2

3/2 -5/2A-1=

6543

Class assignment 2

• Find the matrix A.

• Let A be an invertible matrix whose inverse is:

Don’t overthink this question. Invert this matrix. (There is a reason for this wording, discussed later in the semester.)

Class assignment 2 Solution

• Find the matrix A.

( ) ( )[ ] ( ) ( )

−

−=

−

−

−

=

−=−=−=

=

2/32/523

23546

A

220185x46x3)det(A ,6543

A 1-1-