row- horizontal elements column- vertical elements 3 x 2 matrix

Row- horizontal elementsColumn- vertical elements

3 x 2 matrix

310

221A

412

403B

2BA

If A and B are both m × n matrices then the sum of A and B, denoted A + B, is a matrix obtained by adding corresponding elements of A and B.

add these

310

221A

412

403B

22BA

add these

310

221A

412

403B

622BA

add these

310

221A

412

403B

2

622BA

add these

310

221A

412

403B

02

622BA

add these

310

221A

412

403B

102

622BA

add these

If A and B are both m × n matrices then the difference of A and B, denoted A - B, is a matrix obtained by subtracting corresponding elements of A and B.

ABBA

CBACBA )()(

The multiplication of matrices is easier shown than put into words. You multiply the rows of the first matrix with the columns of the second adding products

140

123A

13

31

42

B

Find AB

First we multiply across the first row and down the first column adding products. We put the answer in the first row, first column of the answer.

23 1223 5311223

140

123A

13

31

42

B

Find AB

We multiplied across first row and down first column so we put the answer in the first row, first column.

5AB

Now we multiply across the first row and down the second column and we’ll put the answer in the first row, second column.

43 3243 7113243

75AB

Now we multiply across the second row and down the first column and we’ll put the answer in the second row, first column.

20 1420 1311420

1

75AB

Now we multiply across the second row and down the second column and we’ll put the answer in the second row, second column.

40 3440 11113440

111

75AB

Notice the sizes of A and B and the size of the product AB.

To multiply matrices A and B look at their dimensions

pnnm MUST BE SAME

SIZE OF PRODUCT

If the number of columns of A does not equal the number of rows of B then the

product AB is undefined.

6

BA

126

BA

2126

BA

3

2126

BA

143

2126

BA

4143

2126

BA

9

4143

2126

BA

109

4143

2126

BA

4109

4143

2126

BA

Now let’s look at the product BA.

13

31

42

B

140

123A

BAAB

23

32

can multiply

size of answer

across first row as we go down first column:

60432

across first row as we go down second column:

124422

across first row as we go down third column:

21412

across second row as we go down first column:

30331

across second row as we go down second column:

144321

across second row as we go down third column:

41311

across third row as we go down first column:

90133

across third row as we go down second column:

104123

across third row as we go down third column:

41113

Completely different than AB!

Commuter's Beware!

13

Ca

K

H He

Li

B

Be

C N O FNe

NaMg

Al Si P S Cl

Ar

0

50

100

150

200

250

0 2 4 6 8 10 12 14 16 18 20

Element

Ato

mic

Rad

ius

(pm

)

Atomic Radii refers to the size of the atom

It is measured from the nucleus to the edge of the atom

What is the trend?The radius

increases as you go down a group.

The radius decreases as you

go across a period

18

period 1

period 2

period 3

1p+

3p+ 4n0

11p+ 12n0

As we move down a group, the valence electron(s) move farther away from the nucleus because the period number tells us what energy level the valence electron is on

Group 1

Li

+++

Be

++++

B

+++++

Period 2

Going ACROSS a period proton # increases . More protons means greater attraction between nucleus and valence electron(s).

The greater attraction also means that valence electrons are brought closer to the nucleus, thus smaller atomic radius results.

Group 1 Group 2 Group 13

Barium or Magnesium? Barium

Sodium

Cesium

• Lithium or Sodium?

• Potassium, Argon, Neon, or Cesium?

21

Ionization energy is the energy required to remove an electron from an atom.

What is the trend?Why?

Decreases down a group

SHIELDING Interior electrons

prevent the nucleus from pulling on valence electrons

1p+

3p+ 4n0

11p+ 12n0

Increases across a period, or row Increased electrostatic attraction

24

Li

+++

Be

++++

B

+++++

Peroid 2

Group 1 Group 2 Group 13

25

Electronegativity is a measure of the ability of an atom in a chemical compound to attract electrons

What is the trend?Why?

Decreases down a group

SHIELDING Interior electrons

prevent the nucleus from pulling on valence electrons

1p+

3p+ 4n0

11p+ 12n0

Increases across a period Increased electrostatic attraction

28

Li

+++

Be

++++

B

+++++

Peroid 2

Group 1 Group 2 Group 13

Atomic Radius: down and left Ionization Energy: up and rightElectronegativity: up and right

WHY? Answer is always shielding down a group

and increased electrostatic attraction across a period

29

Matrices Worksheet (page 4 of the packet)

Encode a 10-character message using the matrix given (due Wed)