MATRIX ALGEBRA
Tutorial Manual
MUT
Maths 1
,Table of Contents
1 Intended Learning Outcomes ................................................................................................. 2
2 Introduction ................................................................................................................................. 2
2.1 Exercise ................................................................................................................................ 3
3 Matrix ............................................................................................................................................. 3
4 Addition and Subtraction of Matrices ................................................................................... 4
5 Multiplication of Matrices ......................................................................................................... 4
5.1 Exercise ................................................................................................................................ 5
5.2 Exercise ................................................................................................................................ 5
6 Determinant of a 2x2 matrix .................................................................................................... 7
6.1 Exercise ................................................................................................................................ 7
6.2 Exercise ................................................................................................................................ 7
6.3 Exercise ................................................................................................................................ 8
7 Minor and cofactor of a matrix ............................................................................................... 8
7.1 Remark:................................................................................................................................. 9
7.2 Remark:............................................................................................................................... 10
8 Determinant of a 3x3 matrix by cofactor expansion ....................................................... 10
8.1 Example .............................................................................................................................. 10
8.1.1 Remark:....................................................................................................................... 11
8.2 Exercise .............................................................................................................................. 12
8.3 Exercise .............................................................................................................................. 12
9 Cramer’s rule ............................................................................................................................. 13
9.1 Solving 2 equations with 2 unknowns using Cramer’s rule ................................. 14
9.1.1 Example ...................................................................................................................... 14
9.2 Solving 3 equations with 3 unknowns using Cramer’s rule ................................. 15
9.2.1 Example ...................................................................................................................... 15
10 Tutorial Questions ............................................................................................................... 16
11 Conclusion and References .............................................................................................. 16
1
, 1 Intended Learning Outcomes
By the end of this handbook, students should be able to
Evaluate the determinant of a 2x2 matrix
Evaluate the inverse of a 2x2 matrix
Evaluate the determinant of a 3x3 matrix using cofactor expansion along the
first, second and third row
Use Cramer’s rule to solve a system of linear equations with 2 equations and
2 unknowns
Use Cramer’s rule to solve a system of linear equations with 3 equations and
3 unknowns
2 Introduction
When two or more equations have the same variables and solutions, we say they are
simultaneous equations. For example, the following are all simultaneous equations:
x y 3
a)
x 2y 5
x y 5
b)
x y2 4
2
2 x 3 y 2
c)
x 2 2 xy 12
2x 3y 7
d)
6x y 1
5 x 6 y 3z 6
e) 4 x 7 y 2 z 3
3x y 7 z 1
A linear equation does not involve any product or powers of variables. For this reason,
(b), (c), and (d) are not linear simultaneous equations. For Maths 1, we are only
interested in systems of linear equations.
2
Tutorial Manual
MUT
Maths 1
,Table of Contents
1 Intended Learning Outcomes ................................................................................................. 2
2 Introduction ................................................................................................................................. 2
2.1 Exercise ................................................................................................................................ 3
3 Matrix ............................................................................................................................................. 3
4 Addition and Subtraction of Matrices ................................................................................... 4
5 Multiplication of Matrices ......................................................................................................... 4
5.1 Exercise ................................................................................................................................ 5
5.2 Exercise ................................................................................................................................ 5
6 Determinant of a 2x2 matrix .................................................................................................... 7
6.1 Exercise ................................................................................................................................ 7
6.2 Exercise ................................................................................................................................ 7
6.3 Exercise ................................................................................................................................ 8
7 Minor and cofactor of a matrix ............................................................................................... 8
7.1 Remark:................................................................................................................................. 9
7.2 Remark:............................................................................................................................... 10
8 Determinant of a 3x3 matrix by cofactor expansion ....................................................... 10
8.1 Example .............................................................................................................................. 10
8.1.1 Remark:....................................................................................................................... 11
8.2 Exercise .............................................................................................................................. 12
8.3 Exercise .............................................................................................................................. 12
9 Cramer’s rule ............................................................................................................................. 13
9.1 Solving 2 equations with 2 unknowns using Cramer’s rule ................................. 14
9.1.1 Example ...................................................................................................................... 14
9.2 Solving 3 equations with 3 unknowns using Cramer’s rule ................................. 15
9.2.1 Example ...................................................................................................................... 15
10 Tutorial Questions ............................................................................................................... 16
11 Conclusion and References .............................................................................................. 16
1
, 1 Intended Learning Outcomes
By the end of this handbook, students should be able to
Evaluate the determinant of a 2x2 matrix
Evaluate the inverse of a 2x2 matrix
Evaluate the determinant of a 3x3 matrix using cofactor expansion along the
first, second and third row
Use Cramer’s rule to solve a system of linear equations with 2 equations and
2 unknowns
Use Cramer’s rule to solve a system of linear equations with 3 equations and
3 unknowns
2 Introduction
When two or more equations have the same variables and solutions, we say they are
simultaneous equations. For example, the following are all simultaneous equations:
x y 3
a)
x 2y 5
x y 5
b)
x y2 4
2
2 x 3 y 2
c)
x 2 2 xy 12
2x 3y 7
d)
6x y 1
5 x 6 y 3z 6
e) 4 x 7 y 2 z 3
3x y 7 z 1
A linear equation does not involve any product or powers of variables. For this reason,
(b), (c), and (d) are not linear simultaneous equations. For Maths 1, we are only
interested in systems of linear equations.
2
