This document discusses different methods for solving simultaneous equations, including: 1) Using matrices to solve 2 or 3 simultaneous equations by writing the equations in matrix form, taking the inverse of the coefficient matrix, and solving for the variables. 2) Using determinants to solve 2 simultaneous equations by calculating the determinants of the coefficients. 3) Using Cramer's rule and Gaussian elimination to solve simultaneous equations. 4) Calculating eigenvalues and eigenvectors, where eigenvalues are solutions to the characteristic equation determined from the characteristic determinant of a matrix. Examples are provided to illustrate these methods.

1. Analytical Mathematics
Lecture 2
ANJULA WEERASINGHE
BENG. EE (SHU.UK), MSC. CS (UOP)

2. Applications of matrices & determinants

3. Solution of simultaneous equations by
matrices – ( 2 unknowns )
 The procedure for solving linear simultaneous equations in two unknowns using
matrices is:
i. write the equations in the form
ii. Write the matrix equation corresponding to these equations,

4. iii. Determine the inverse matrix of
iv. multiply each side of (ii) by the inverse matrix , and
v. Solve for x and y by equating corresponding elements.

5. Example
1. Use matrices to solve the simultaneous equations:

6. Solution of simultaneous equations by
matrices – ( 3 unknowns )
 The procedure for solving linear simultaneous equations in three unknowns using
matrices is:
i. write the equations in the form
ii. Write the matrix equation corresponding to these equations, i.e.

7. iii. Determine the inverse matrix of
iv. multiply each side of (ii) by the inverse matrix , and
v. Solve for x , y and z by equating corresponding elements.

8. Example
1. Use matrices to solve the simultaneous equations:

9. Solution of simultaneous equations by
determinants
 When solving linear simultaneous equations in two unknowns using
determinants:
i. Write the equations in the form
ii. The solution is given by

10. Where
i.e. the determinant of the coefficients left when the x-column is covered up,
i.e. the determinant of the coefficients left when the y-column is covered up,
And
i.e. the determinant of the coefficients left when the constants -column is covered
up,

11. Example
1. Solve the following simultaneous equations using determinants:

12. Example
1. The velocity of a car, accelerating at uniform acceleration a between two points, is
given by v=u+at, where u is its velocity when passing the first point and t is the
time taken to pass between the two points.
If v=21m/s when t =3.5s and v=33m/s when t =6.1s, use determinants to
find the values of u and a, each correct to 4 significant figures.

13. Solution of simultaneous 3 equations by
determinants
 When solving simultaneous equations in three unknowns using determinants:
i. Write the equations in the form
ii. The solution is given by

14. Where
i.e. the determinant of the coefficients left when the x-column is covered up,
i.e. the determinant of the coefficients left when the y-column is covered up,
And
i.e. the determinant of the coefficients left when the constants -column is covered
up,

15. Example
1. A d.c. circuit comprises three closed loops. Applying Kirchhoff’s laws to the closed
loops gives the following equations for current flow in milliamperes:
Use determinants to solve for and

16. Solution of simultaneous equations using
Cramer’s rule
 Cramer’s∗ rule states that if

17. Where
i.e. the x-column has been replaced by the RHS b column ,
i.e. the y-column has been replaced by the RHS b column ,
i.e. the z-column has been replaced by the RHS b column ,
,

18. Example
1. Solve the following simultaneous equations using Cramer’s rule.

19. Solution of simultaneous equations using the
Gaussian elimination method
 Consider the following simultaneous equations:

23. Example
 A d.c. circuit comprises three closed loops. Applying Kirchhoff’s laws to the closed
loops gives the following equations for current flow in milliamperes:
Use the Gaussian elimination method to solve for I1, I2 and I3

24. Eigenvalues and Eigenvectors
 In practical applications, such as coupled oscillations and vibrations, equations of the form:
 occur, where A is a square matrix and λ is a number. Whenever x ≠ 0, the values of λ are called
the eigenvalues of the matrix A; the corresponding solutions of the equation Ax = λx are
called the eigenvectors of A.

25.  Sometimes, instead of the term eigenvalues, characteristic values or latent
roots are used. Also, instead of the term eigenvectors, characteristic vectors is
used. From above, if Ax = λx then Ax −λx = 0 i.e.
(A −λI) = 0 where I is the unit matrix.
If x ≠ 0 then |A−λI| = 0
 |A−λI| is called the characteristic determinant of A and |A−λI| = 0 is called the
characteristic equation.

26. Example
1. Determine the eigenvalues of the matrix
2. Determine the eigenvectors of the matrix

27. Example
3. Determine the eigenvalues of the matrix
4. Determine the eigenvectors of the matrix

28. Example
3. Determine the eigenvalues of the matrix
4. Determine the eigenvectors of the matrix