Determinants of 2 x 2 and 3 x 3 Matrices

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Presentation transcript:

Determinants of 2 x 2 and 3 x 3 Matrices

Determinant of a 2 x 2 Matrix b c d d b a c -

Example 1

Determinant of a 2 x 2 Matrix 3 5 4 6 3 6 4 5 - -2 =

Example 2

Determinant of a 2 x 2 Matrix -4 3 5 2 -4 2 5 3 - -23 =

Example 3

Determinant of a 2 x 2 Matrix 8 4 6 5 8 5 6 4 - 16 =

Example 4

Determinant of a 3 x 3 Matrix 1 3 4 2 5 6 7

Determinant of a 3 x 3 Matrix 1 3 4 2 5 6 7 Augment the determinant with the first two columns

Determinant of a 3 x 3 Matrix 1 3 4 2 5 6 7 Augment the determinant with the first two columns

Determinant of a 3 x 3 Matrix 1 3 4 2 5 6 7 Augment the determinant with the first two columns

Determinant of a 3 x 3 Matrix 1 3 4 2 5 6 7 Augment the determinant with the first two columns

Determinant of a 3 x 3 Matrix 1 3 4 2 5 6 7

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - - -

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - - -

Determinant of a 3 x 3 Matrix 1 3 4 2 5 6 7 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix 1 3 4 2 5 6 7 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + 1 3 4 2 5 6 7 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + 1 3 4 2 5 6 7 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - - 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - - 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - - - 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - - - 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - - - 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - - - 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix + + 1 3 4 2 5 6 7 - - - 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3 7 + 45 + 48 - 12 - 30 - 42 = 16

Example 5

Determinant of a 3 x 3 Matrix 2 1 6 4 3 7 5 9 8 Augment the determinant with the first two columns

Determinant of a 3 x 3 Matrix 2 1 6 4 3 7 5 9 8 Augment the determinant with the first two columns

Determinant of a 3 x 3 Matrix 2 1 6 4 3 7 5 9 8 Augment the determinant with the first two columns

Determinant of a 3 x 3 Matrix 2 1 6 4 3 7 5 9 8 Augment the determinant with the first two columns

Determinant of a 3 x 3 Matrix 2 1 6 4 3 7 5 9 8 Augment the determinant with the first two columns

Determinant of a 3 x 3 Matrix 2 1 6 4 3 7 5 9 8 Augment the determinant with the first two columns

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - - -

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - - -

Determinant of a 3 x 3 Matrix 2 1 6 4 3 7 5 9 8 1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3

Determinant of a 3 x 3 Matrix 2 1 6 4 3 7 5 9 8 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + 2 1 6 4 3 7 5 9 8 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + 2 1 6 4 3 7 5 9 8 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - - 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - - 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - - - 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - - - 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - - - 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - - - 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1

Determinant of a 3 x 3 Matrix + + 2 1 6 4 3 7 5 9 8 - - - 2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1 48 + 35 + 216 - 90 - 126 - 32 = 51

Find the area of the Triangle (5, 9) Use a Determinant (9, 5) 1 2 5 9 (1, 2)

Determinant of a 3 x 3 Matrix + + 1 2 5 9 - - - (1•9•1 + 2•1•9 + 1•5•5 - 9•9•1 - 5•1•1 - 1•5•2) (9 + 18 + 25 - 81 - 5 - 10) 22

Find the area of the Quadrilateral (3, 9) (9, 5) (-3, 2) Divide into two triangles (1, -5)

Find the area of the Quadrilateral (3, 9) The top triangle (9, 5) (-3, 2) -3 2 1 3 9 5 (1, -5)

Find the area of the Quadrilateral (3, 9) The bottom triangle (9, 5) (-3, 2) 9 5 1 -5 -3 2 (1, -5)

Find the area of the Quadrilateral (3, 9) (9, 5) (-3, 2) Purple Area = Green Area = (1, -5) Total Area =

The area of the triangle is 11. Find the value(s) of k. Use a Determinant (1, 3) k+1 1 k 5 3 (k+1, 1)

The area of the triangle is 11. Find the value(s) of k. 5 3

The area of the triangle is 11. Find the value(s) of k. 5 3

“Shoe Lace” Method

Find the area of the Triangle (5, 9) 1 2 5 9 (9, 5) (1, 2)

Find the area of the Quadrilateral (3, 9) -3 2 3 9 5 1 -5 (9, 5) (-3, 2) (1, -5)