Section 4.3 – A Review of DeterminantsSection 4.4 – The Cross Product

Vocabulary First (Again )

Determinant – a number (scalar)

1 2 3

det A or 4 5 6

7 8 9

Notations

The 2 x 2 Determinant

a b

c dad bc

7 1

2 3 7 3 2 1 19

3 4

1 2 3 2 1 4 10

6 2

0 3

1 4

2 5

2 1

7 6

7 3

0 0

18 0 18

5 8 13

12 7 5

0 0 0

Try these four… …and these four

13

22 6

1 4

2 3

1 5

6 2

3 0

0 5

3 6 3

3 8 5

2 30 28

15 0 15

The MINOR of a matrix 1. Cross out the row and column of the element2. Compute the determinant of what remains

5

2 3

4

1

6

7 8 9

15 6

The minor of is or 38 9

24 6

The minor of is or 67 9

34 5

The minor of is or 37 8

51 3

The minor of is or 127 9

The 3 x 3 Determinant

1. Select ANY row or column (most zeros would be smart)

2. Take each element and multiply it by its MINOR.

3. Apply + - + - + - (to be explained). Remember the + starts with the first row first column element.

3 1 0

2 3 5

1 2 1

3 53

2 1

2 51

1 1

2 30

1 2+ – +

21 3 0 18

3 1 0

2 3 5

1 2 1

2 30

1 2

3 15

1 2

3 11

2 3+ – +

0 25 7 18

3 1 0

2 3 5

1 2 1

2 51

1 1

3 03

1 1

3 02

2 5– + –

3 9 30 18

3 1 0

2 3 5

1 2 1

1 02

2 1

3 03

1 1

3 15

1 2– + –

2 9 25 18

– + –

2 5 1

0 0 7

4 2 30 0

2 57

4 2

7 16 112

2 1 3

4 2 1

3 2 1

1 33

2 12 3

24 1

2 11

4 2+ – +

21 20 8 7

3 1 7

0 0 0

5 2 3= 0

Definition

The cross product of two vector yields a vector whichIs orthogonal to the two given vectors.

If A = ai + bj + ck and B = di + ej + fk

i j k

A B a b c

d e f

a b a c a b

d e d f ej k

di

Find the vector orthogonal to A = 2i + 3j + k and B = 3i - 2j + 5k

2 3 1

i j k

3 2 5

3 1 2 1 2 3

2 5 3 5 2i j k

3

i j17 7 3k1

Find the vector orthogonal to A = 7i + 1j + 2k and B = i + 3j + 4k

7 1

i j k

2

1 3 4

1 2 7 2 7 1

3 4 1 4j k

3i

1

2 26 0i 2j k