11 x1 t01 10 matrices (2012)
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Using Matrices to Solve Simultaneous Equations
Using Matrices to Solve Simultaneous Equations
2 2 matrix:a b
Ac d
Using Matrices to Solve Simultaneous Equations
2 2 matrix:a b
Ac d
Determinant of A: det A A ad bc
Using Matrices to Solve Simultaneous Equations
2 2 matrix:a b
Ac d
Determinant of A: det A A ad bc
Adjoint matrix: adjd b
Ac a
Using Matrices to Solve Simultaneous Equations
2 2 matrix:a b
Ac d
Determinant of A: det A A ad bc
Adjoint matrix: adjd b
Ac a
a bc d
swap
Using Matrices to Solve Simultaneous Equations
2 2 matrix:a b
Ac d
Determinant of A: det A A ad bc
Adjoint matrix: adjd b
Ac a
a bc d
swap
change signs
Using Matrices to Solve Simultaneous Equations
2 2 matrix:a b
Ac d
Determinant of A: det A A ad bc
Inverse of A:
Adjoint matrix: adjd b
Ac a
a bc d
swap
change signs
1 adjAAA
Using Matrices to Solve Simultaneous Equations
2 2 matrix:a b
Ac d
Determinant of A: det A A ad bc
Inverse of A:x m
Ay n
1x mA
y n
Adjoint matrix: adjd b
Ac a
a bc d
swap
change signs
1 adjAAA
Using Matrices to Solve Simultaneous Equations
2 2 matrix:a b
Ac d
Determinant of A: det A A ad bc
Inverse of A:x m
Ay n
1x mA
y n
Adjoint matrix: adjd b
Ac a
a bc d
swap
change signs
1 adjAAA
If 0, then lines are paralleli.e. no solutions
A
e.g. ( ) 2 3 21 (1)5 2 3 (2)
i x yx y
e.g. ( ) 2 3 21 (1)5 2 3 (2)
i x yx y
2 3 215 2 3
xy
e.g. ( ) 2 3 21 (1)5 2 3 (2)
i x yx y
2 3 215 2 3
xy
2 2 5 3A 2 53 2
e.g. ( ) 2 3 21 (1)5 2 3 (2)
i x yx y
2 3 215 2 3
xy
2 2 5 3A 2 53 2
2adj
2A
1. swap
2 3adj
5 2A
e.g. ( ) 2 3 21 (1)5 2 3 (2)
i x yx y
2 3 215 2 3
xy
2 2 5 3A 2 53 2
1. swap
2.change signs
2 3adj
5 2A
e.g. ( ) 2 3 21 (1)5 2 3 (2)
i x yx y
2 3 215 2 3
xy
2 2 5 3A
2 3 21111 5 2 3
xy
2 53 2
1. swap
2.change signs
2 3adj
5 2A
e.g. ( ) 2 3 21 (1)5 2 3 (2)
i x yx y
2 3 215 2 3
xy
2 2 5 3A
2 3 21111 5 2 3
xy
Multiply the row of the matrix with the column
of the vector
2 21 3 3111 5 21 2 3
xy
33111 99
xy
2 53 2
1. swap
2.change signs
2 3adj
5 2A
e.g. ( ) 2 3 21 (1)5 2 3 (2)
i x yx y
2 3 215 2 3
xy
2 2 5 3A
2 3 21111 5 2 3
xy
Multiply the row of the matrix with the column
of the vector
2 21 3 3111 5 21 2 3
xy
33111 99
xy
39
xy
2 53 2
1. swap
2.change signs
2 3adj
5 2A
e.g. ( ) 2 3 21 (1)5 2 3 (2)
i x yx y
2 3 215 2 3
xy
2 2 5 3A
2 3 21111 5 2 3
xy
Multiply the row of the matrix with the column
of the vector
3, 9x y
2 21 3 3111 5 21 2 3
xy
33111 99
xy
39
xy
2 53 2
1. swap
2.change signs
( ) 4 5 3 (1)3 7 13 (2)
ii x yx y
( ) 4 5 3 (1)3 7 13 (2)
ii x yx y
4 5 33 7 13
xy
7 5 31
43 3 4 13xy
( ) 4 5 3 (1)3 7 13 (2)
ii x yx y
4 5 33 7 13
xy
7 5 31
43 3 4 13xy
7 3 5 13143 3 3 4 13
xy
86143 43
xy
( ) 4 5 3 (1)3 7 13 (2)
ii x yx y
4 5 33 7 13
xy
7 5 31
43 3 4 13xy
2, 1x y
7 3 5 13143 3 3 4 13
xy
21
xy
86143 43
xy
3 3 matrix:a b c
A d e fg h i
3 3 matrix:a b c
A d e fg h i
Determinant of A: det A A aei bfg cdh gec hfa idb
3 3 matrix:a b c
A d e fg h i
Determinant of A: det A A aei bfg cdh gec hfa idb
Adjoint matrix: adj
te f d f d eh i g i g hb c a c a b
Ah i g i g h
b c a c a be f d f d e
3 3 matrix:a b c
A d e fg h i
Determinant of A: det A A aei bfg cdh gec hfa idb
Adjoint matrix: adj
te f d f d eh i g i g hb c a c a b
Ah i g i g h
b c a c a be f d f d e
a b cd e fg h i
1. cover up row and column of the position looking for
3 3 matrix:a b c
A d e fg h i
Determinant of A: det A A aei bfg cdh gec hfa idb
Adjoint matrix: adj
te f d f d eh i g i g hb c a c a b
Ah i g i g h
b c a c a be f d f d e
a b cd e fg h i
1. cover up row and column of the position looking for
2. find determinant of the remaining matrix
3 3 matrix:a b c
A d e fg h i
Determinant of A: det A A aei bfg cdh gec hfa idb
Adjoint matrix: adj
te f d f d eh i g i g hb c a c a b
Ah i g i g h
b c a c a be f d f d e
t = transpose rows and columns
a b cd e fg h i
1. cover up row and column of the position looking for
2. find determinant of the remaining matrix
e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)
x y zx y zx y z
e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)
x y zx y zx y z
1 2 1 52 3 4 284 5 3 10
xyz
e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)
x y zx y zx y z
1 2 1 52 3 4 284 5 3 10
xyz
1 2 12 3 44 5 3
A
e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)
x y zx y zx y z
1 2 1 52 3 4 284 5 3 10
xyz
1 2 12 3 44 5 3
A
1 2 1 1 22 3 4 2 34 5 3 4 5
e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)
x y zx y zx y z
1 2 1 52 3 4 284 5 3 10
xyz
1 2 12 3 44 5 3
A
1 2 1 1 22 3 4 2 34 5 3 4 5
1 3 3 2 4 4 1 2 5
9 32 10
e.g 2 5 (1)2 3 4 28 (2)4 5 3 10 (3)
x y zx y zx y z
1 2 1 52 3 4 284 5 3 10
xyz
1 2 12 3 44 5 3
A
1 2 1 1 22 3 4 2 34 5 3 4 5
1 3 3 2 4 4 1 2 5
9 32 10 4 3 1 5 4 1 3 2 2
12 20 12 11
3 4 2 4 2 35 3 4 3 4 52 1 1 1 1 2
adj5 3 4 3 4 52 1 1 1 1 23 4 2 4 2 3
t
A
1 2 12 3 44 5 3
A
3 4 2 4 2 35 3 4 3 4 52 1 1 1 1 2
adj5 3 4 3 4 52 1 1 1 1 23 4 2 4 2 3
t
A
1 2 12 3 44 5 3
A
11 22 221 1 35 6 7
t
3 4 2 4 2 35 3 4 3 4 52 1 1 1 1 2
adj5 3 4 3 4 52 1 1 1 1 23 4 2 4 2 3
t
A
1 2 12 3 44 5 3
A
11 22 221 1 35 6 7
t
11 1 522 1 622 3 7
1 2 1 52 3 4 284 5 3 10
xyz
1 2 1 52 3 4 284 5 3 10
xyz
11 1 5 51 22 1 6 28
1122 3 7 10
xyz
1 2 1 52 3 4 284 5 3 10
xyz
11 1 5 51 22 1 6 28
1122 3 7 10
xyz
11 5 1 28 5 101 22 5 1 28 6 10
1122 5 3 28 7 10
xyz
331 22
1144
xyz
1 2 1 52 3 4 284 5 3 10
xyz
11 1 5 51 22 1 6 28
1122 3 7 10
xyz
11 5 1 28 5 101 22 5 1 28 6 10
1122 5 3 28 7 10
xyz
331 22
1144
xyz
32
4
xyz
3, 2, 4x y z
1 2 1 52 3 4 284 5 3 10
xyz
11 1 5 51 22 1 6 28
1122 3 7 10
xyz
11 5 1 28 5 101 22 5 1 28 6 10
1122 5 3 28 7 10
xyz
331 22
1144
xyz
32
4
xyz
3, 2, 4x y z
Worksheet+
try some ofExercise 1H;
1, 2, 6