Download - 11X1 T01 10 matrices
Using Matrices to Solve
Simultaneous Equations
Using Matrices to Solve
Simultaneous Equations
11 12
21 22
a aA
a a
2 2 matrix
Using Matrices to Solve
Simultaneous Equations
11 12
21 22
a aA
a a
11 22 21 12det A A a a a a
2 2 matrix Determinant of A
Using Matrices to Solve
Simultaneous Equations
11 12
21 22
a aA
a a
11 22 21 12det A A a a a a
2 2 matrix Determinant of A
Inverse of A
22 121
21 11
1 a aA
a aA
Using Matrices to Solve
Simultaneous Equations
11 12
21 22
a aA
a a
11 22 21 12det A A a a a a
2 2 matrix Determinant of A
Inverse of A
22 121
21 11
1 a aA
a aA
x mA
y n
Solving simultaneous
equations
1x m
Ay n
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
2 3 21
5 2 3
x
y
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
2 3 21
5 2 3
x
y
2 3 211
11 5 2 3
x
y
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
2 3 21
5 2 3
x
y
2 2 5 3A
2 3 211
11 5 2 3
x
y
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
2 3 21
5 2 3
x
y
2 2 5 3A
2 3 211
11 5 2 3
x
y
swap
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
2 3 21
5 2 3
x
y
2 2 5 3A
2 3 211
11 5 2 3
x
y
swap
change
signs
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
2 3 21
5 2 3
x
y
2 2 5 3A
2 3 211
11 5 2 3
x
y
swap
change
signs 2 21 3 31
11 5 21 2 3
x
y
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
2 3 21
5 2 3
x
y
2 2 5 3A
2 3 211
11 5 2 3
x
y
Multiply the row of the
matrix with the column
of the vector
swap
change
signs 2 21 3 31
11 5 21 2 3
x
y
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
2 3 21
5 2 3
x
y
2 2 5 3A
2 3 211
11 5 2 3
x
y
Multiply the row of the
matrix with the column
of the vector
swap
change
signs 2 21 3 31
11 5 21 2 3
x
y
331
11 99
x
y
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
2 3 21
5 2 3
x
y
2 2 5 3A
2 3 211
11 5 2 3
x
y
Multiply the row of the
matrix with the column
of the vector
swap
change
signs 2 21 3 31
11 5 21 2 3
x
y
331
11 99
x
y
3
9
x
y
e.g. ( ) 2 3 21 (1)
5 2 3 (2)
i x y
x y
2 3 21
5 2 3
x
y
2 2 5 3A
2 3 211
11 5 2 3
x
y
Multiply the row of the
matrix with the column
of the vector
3, 9x y
swap
change
signs 2 21 3 31
11 5 21 2 3
x
y
331
11 99
x
y
3
9
x
y
( ) 171 213 642 (1)
114 326 244 (2)
ii x y
x y
( ) 171 213 642 (1)
114 326 244 (2)
ii x y
x y
171 213 642
114 326 244
x
y
( ) 171 213 642 (1)
114 326 244 (2)
ii x y
x y
171 213 642
114 326 244
x
y
326 213 6421
31464 114 171 244
x
y
( ) 171 213 642 (1)
114 326 244 (2)
ii x y
x y
171 213 642
114 326 244
x
y
326 213 6421
31464 114 171 244
x
y
1573201
31464 31464
x
y
( ) 171 213 642 (1)
114 326 244 (2)
ii x y
x y
171 213 642
114 326 244
x
y
326 213 6421
31464 114 171 244
x
y
1573201
31464 31464
x
y
5
1
x
y
( ) 171 213 642 (1)
114 326 244 (2)
ii x y
x y
171 213 642
114 326 244
x
y
326 213 6421
31464 114 171 244
x
y
5, 1x y
1573201
31464 31464
x
y
5
1
x
y
11 1
1
n
n nn
a a
A
a a
matrixn n
11 1
1
n
n nn
a a
A
a a
1
1 1
1
1n
k
k k
k
A a M
matrixn n
1where is the matrix with
column 1 and row removed
kM
k
11 1
1
n
n nn
a a
A
a a
1
1 1
1
1n
k
k k
k
A a M
matrixn n
1where is the matrix with
column 1 and row removed
kM
k
3 2 1
e.g. ( ) 1 4 2
0 1 2
i
11 1
1
n
n nn
a a
A
a a
1
1 1
1
1n
k
k k
k
A a M
matrixn n
1where is the matrix with
column 1 and row removed
kM
k
3 2 1
e.g. ( ) 1 4 2
0 1 2
i4 2 1 2 1 4
3 21 2 0 2 0 1
11 1
1
n
n nn
a a
A
a a
1
1 1
1
1n
k
k k
k
A a M
matrixn n
1where is the matrix with
column 1 and row removed
kM
k
3 2 1
e.g. ( ) 1 4 2
0 1 2
i4 2 1 2 1 4
3 21 2 0 2 0 1
3 6 2 2 1
15
1 3 5 2
0 1 3 4( )
2 1 9 6
3 2 4 8
ii
1 3 5 2
0 1 3 4( )
2 1 9 6
3 2 4 8
ii
1 3 4 0 3 4 0 1 4 0 1 3
1 9 6 3 2 9 6 5 2 1 6 2 2 1 9
2 4 8 3 4 8 3 2 8 3 2 4
1 3 5 2
0 1 3 4( )
2 1 9 6
3 2 4 8
ii
1 3 4 0 3 4 0 1 4 0 1 3
1 9 6 3 2 9 6 5 2 1 6 2 2 1 9
2 4 8 3 4 8 3 2 8 3 2 4
9 6 1 6 1 9 9 6 2 6 2 93 4 3 0 3 4
4 8 2 8 2 4 4 8 3 8 3 4
1 6 2 6 2 1 1 9 2 9 2 15 0 4 2 0 3
2 8 3 8 3 2 2 4 3 4 3 2
1 3 5 2
0 1 3 4( )
2 1 9 6
3 2 4 8
ii
1 3 4 0 3 4 0 1 4 0 1 3
1 9 6 3 2 9 6 5 2 1 6 2 2 1 9
2 4 8 3 4 8 3 2 8 3 2 4
9 6 1 6 1 9 9 6 2 6 2 93 4 3 0 3 4
4 8 2 8 2 4 4 8 3 8 3 4
1 6 2 6 2 1 1 9 2 9 2 15 0 4 2 0 3
2 8 3 8 3 2 2 4 3 4 3 2
4 3 4 4 11 3 0 3 2 4 19 5 0 2 4 1
2 0 19 3 1
1 3 5 2
0 1 3 4( )
2 1 9 6
3 2 4 8
ii
1 3 4 0 3 4 0 1 4 0 1 3
1 9 6 3 2 9 6 5 2 1 6 2 2 1 9
2 4 8 3 4 8 3 2 8 3 2 4
9 6 1 6 1 9 9 6 2 6 2 93 4 3 0 3 4
4 8 2 8 2 4 4 8 3 8 3 4
1 6 2 6 2 1 1 9 2 9 2 15 0 4 2 0 3
2 8 3 8 3 2 2 4 3 4 3 2
4 3 4 4 11 3 0 3 2 4 19 5 0 2 4 1
2 0 19 3 1
160
or convert to a triangular matrix (create a triangle of 0’s)
3 2 1
e.g. ( ) 1 4 2
0 1 2
i
or convert to a triangular matrix (create a triangle of 0’s)
3 2 1
e.g. ( ) 1 4 2
0 1 2
i
3 2 1
10 50
3 3
0 1 2
1
2 13
row row
or convert to a triangular matrix (create a triangle of 0’s)
3 2 1
e.g. ( ) 1 4 2
0 1 2
i
3 2 1
10 50
3 3
0 1 2
3 2 1
10 50
3 3
30 0
2
3
3 210
row row
12 1
3row row
or convert to a triangular matrix (create a triangle of 0’s)
3 2 1
e.g. ( ) 1 4 2
0 1 2
i
3 2 1
10 50
3 3
0 1 2
3 2 1
10 50
3 3
30 0
2
10 33
3 2 multiply the diagonal
12 1
3row row
33 2
10row row
or convert to a triangular matrix (create a triangle of 0’s)
3 2 1
e.g. ( ) 1 4 2
0 1 2
i
3 2 1
10 50
3 3
0 1 2
3 2 1
10 50
3 3
30 0
2
10 33
3 2 multiply the diagonal
15
12 1
3row row
33 2
10row row
1 3 5 2
0 1 3 4( )
2 1 9 6
3 2 4 8
ii
1 3 5 2
0 1 3 4( )
2 1 9 6
3 2 4 8
ii
1 3 5 2
0 1 3 4
0 5 1 2
0 7 11 2
3 2 1row row
4 3 1row row
1 3 5 2
0 1 3 4( )
2 1 9 6
3 2 4 8
ii
1 3 5 2
0 1 3 4
0 5 1 2
0 7 11 2
3 2 1row row
4 3 1row row
1 3 5 2
0 1 3 4
0 0 16 18
0 0 32 26
3 5 2row row
4 7 2row row
1 3 5 2
0 1 3 4( )
2 1 9 6
3 2 4 8
ii
1 3 5 2
0 1 3 4
0 5 1 2
0 7 11 2
3 2 1row row
4 3 1row row
1 3 5 2
0 1 3 4
0 0 16 18
0 0 32 26
3 5 2row row
4 7 2row row
1 3 5 2
0 1 3 4
0 0 16 18
0 0 0 10
4 2 3row row
1 3 5 2
0 1 3 4( )
2 1 9 6
3 2 4 8
ii
1 3 5 2
0 1 3 4
0 5 1 2
0 7 11 2
3 2 1row row
4 3 1row row
1 3 5 2
0 1 3 4
0 0 16 18
0 0 32 26
3 5 2row row
4 7 2row row
1 3 5 2
0 1 3 4
0 0 16 18
0 0 0 10
4 2 3row row
160
1
11 1
1 1
1
( 1)
adj
( 1) ( 1)
tn
n
n n nn
n nn
M M
A
M M
Adjoint matrix
1
11 1
1 1
1
( 1)
adj
( 1) ( 1)
tn
n
n n nn
n nn
M M
A
M M
Adjoint matrix
i.e signs alternate
1
11 1
1 1
1
( 1)
adj
( 1) ( 1)
tn
n
n n nn
n nn
M M
A
M M
Adjoint matrix
i.e signs alternate
t means transpose rows and columns
1
11 1
1 1
1
( 1)
adj
( 1) ( 1)
tn
n
n n nn
n nn
M M
A
M M
Adjoint matrix
i.e signs alternate
t means transpose rows and columns
1 adjAA
A
1
11 1
1 1
1
( 1)
adj
( 1) ( 1)
tn
n
n n nn
n nn
M M
A
M M
Adjoint matrix
i.e signs alternate
t means transpose rows and columns
1 adjAA
A
e.g 2 5 (1)
2 3 4 28 (2)
4 5 3 10 (3)
x y z
x y z
x y z
1 2 1
2 3 4
4 5 3
A
1 2 1
2 3 4
4 5 3
A
1 2 1
2 3 4
4 5 3
A
1 2 1
2 3 4
4 5 3
A
1 2 1
2 3 4
4 5 3
A
1 2 1
0 7 6
0 3 1
1 2 1
2 3 4
4 5 3
A
1 2 1
2 3 4
4 5 3
A
1 2 1
0 7 6
0 3 1
1 2 1
0 7 6
110 0
7
1 2 1
2 3 4
4 5 3
A
1 2 1
2 3 4
4 5 3
A
1 2 1
0 7 6
0 3 1
1 2 1
0 7 6
110 0
7
11
1 2 1
2 3 4
4 5 3
A
1 2 1
2 3 4
4 5 3
A
1 2 1
0 7 6
0 3 1
1 2 1
0 7 6
110 0
7
11
3 4 2 4 2 3
5 3 4 3 4 5
2 1 1 1 1 2adj
5 3 4 3 4 5
2 1 1 1 1 2
3 4 2 4 2 3
t
A
1 2 1
2 3 4
4 5 3
A
1 2 1
2 3 4
4 5 3
A
1 2 1
0 7 6
0 3 1
1 2 1
0 7 6
110 0
7
11
3 4 2 4 2 3
5 3 4 3 4 5
2 1 1 1 1 2adj
5 3 4 3 4 5
2 1 1 1 1 2
3 4 2 4 2 3
t
A
11 1 5
22 1 6
22 3 7
1 2 1 5
2 3 4 28
4 5 3 10
x
y
z
1 2 1 5
2 3 4 28
4 5 3 10
x
y
z
11 1 5 51
22 1 6 2811
22 3 7 10
x
y
z
1 2 1 5
2 3 4 28
4 5 3 10
x
y
z
11 1 5 51
22 1 6 2811
22 3 7 10
x
y
z
3
2
4
x
y
z
1 2 1 5
2 3 4 28
4 5 3 10
x
y
z
11 1 5 51
22 1 6 2811
22 3 7 10
x
y
z
3
2
4
x
y
z
3, 2, 4x y z
1 2 1 5
2 3 4 28
4 5 3 10
x
y
z
11 1 5 51
22 1 6 2811
22 3 7 10
x
y
z
3
2
4
x
y
z
3, 2, 4x y z
Exercise 1H; 1, 2, 6
