Linear Algebra: Simultaneous Equations in Multiple Variables
Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE, Anyone else who needs this Tutorial as a reference.
Linear Algebra - Simultaneous Equations in Multiple Variables - Outline of Contents:
|Representing a system of linear equations in multiple variables in matrix form. Using determinants to solve these systems of equations. Meaning of consistent, homogeneous and non-homogeneous systems of equations. Theorems relating to consistency of systems of equations. Application of Cramer rule. Solved problems demonstrating how to solve linear equations using matrix and determinant related methods. |
Introducing the method to solve systems of linear equations using the determinant method
We introduce the idea of solving simultaneous equations using determinants.
1. Solution: A set of values of the variable which simultaneously satisfy all equations is called a solution of the system of equations
2. Consistent system: If the system of equations has one or more solutions, then it is said to be a consistent system of equations, otherwise it is an inconsistent system of equations.
3. Homogeneous and non-homogeneous system of linear equations: A system of equations is called a homogeneous system if.Otherwise, it is called a non- homogeneous system of equations.
Identifying what kind of solutions a system of equations has based on the Determinants
1. When all the determinants (corresponding to each variable) : In such case the system of equations has a unique solution which is given by Crammer’s rule
2. When the main D = 0
but at least one of the determinants corresponding to one of the variable is non-zero: In this case the system is inconsistent and it has no solutions. The system of equations is non-homogeneous.
3. When all the computed determinants are zero: In this case the system has infinite solutions. The system of equation is homogenous. This system has the trivial solution and and other non-trivial solutions.
Criteria for consistency
When the system is consistent and has a unique solution
When the system is consistent and has infinitely many solutions
When the system is inconsistent
Other topics covered in this tutorial :
Matrix method, Rank method, Solving a non-homogeneous system of linear equations:
Here are the kind of problems solved in this tutorial :
- The sum of three numbers is 6. If we multiply the third number by 3 and add second number to it, we get 11. By adding first and third numbers, we get double of the second number. Represent it algebraically and find the numbers using matrix method.
- Solving a given system of equations
- Testing for the consistency of a system of equations.
- Applying Cramer's rule
- Showing that there is no solution for a given system of equations.
- Finding the conditions under which a homogeneous system of equations possess a non-trivial solution.
Complete Tutorial (MCQ Quiz):
Recap of important aspects of the Matrix Method and Rank method.
Once you're done with the tutorial, you might find this quick summary useful.
Let Ax=B be a system of linear equations with unknowns. If A is non-singular, then A-1 A-1B.
exists and the solution of this system is given by
Else, if A is singular, there might either be infinitely many solution (if (adj A)B =0), or the system may be inconsistent (no solution) (if (adj A)B does not equal 0).
If the system is consistent and has infinitely many solutions, put x3 = k (any real number) and take any 2 equations out of the three. Solve these equations for x1 and x2, which you now obtain, are functions of the third.
1) We explain how to construct the augmented matrix K.
2) The system of equations is consistent if the rank of K = rank of A.
3) If you are provided a system of m simultaneous linear equations in n unknowns.
- if rank A = rank K = n then the system has a unique solution.
- if (rank A = rank K = r) < n, then system of linear equations is consistent and has an infinite number of solutions. In fact, in this case (n-r) variables can be assigned arbitrary values.
- if rank A does not equal rank K then the system is inconsistent (has no solutions)
if (m<n) and rank(A) = rank(K) then r<= m < n and there are infinite solutions
Solving a non-homogenous system of linear equations by rank method
Step 1: Obtain A, B
Step 2: Write augmented matrix K
Step 3: Reduce the augmented matrix to Echelon form by applying a sequence of elementary
Step 4: Determine the number of non-zero rows in and to determine the ranks and
Step 5: If then write the system is inconsistent.
Step 6: If rank(A) = rank(K)
=number of unknowns, then the system has unique solution
which can be obtained by back substitution.
If rank(A) = rank(K) <
number of unknowns, then the system has an infinite number of
solutions which can also be obtained by back substitution.
MCQ Quiz- Simultaneous Equations, Matrices and Determinants- The Basics
A companion MCQ Quiz- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.
You might like to take a look at some of our other Linear Algebra tutorials :
| Introduction to Matrices - Part I Introduction to Matrices. Theory, definitions. What a Matrix is, order of a matrix, equality of matrices, different kind of matrices: row matrix, column matrix, square matrix, diagonal, identity and triangular matrices. Definitions of Trace, Minor, Cofactors, Adjoint, Inverse, Transpose of a matrix. Addition, subtraction, scalar multiplication, multiplication of matrices. Defining special types of matrices like Symmetric, Skew Symmetric, Idempotent, Involuntary, Nil-potent, Singular, Non-Singular, Unitary matrices.||Introduction to Matrices - Part II Problems and solved examples based on the sub-topics mentioned above. Some of the problems in this part demonstrate finding the rank, inverse or characteristic equations of matrices. Representing real life problems in matrix form.||Determinants Introduction to determinants. Second and third order determinants, minors and co-factors. Properties of determinants and how it remains altered or unaltered based on simple transformations is matrices. Expanding the determinant. Solved problems related to determinants. || Simultaneous linear equations in multiple variablesRepresenting a system of linear equations in multiple variables in matrix form. Using determinants to solve these systems of equations. Meaning of consistent, homogeneous and non-homogeneous systems of equations. Theorems relating to consistency of systems of equations. Application of Cramer rule. Solved problems demonstrating how to solve linear equations using matrix and determinant related methods. |
|Basic concepts in Linear Algebra and Vector spacesTheory and definitions. Closure, commutative, associative, distributive laws. Defining Vector space, subspaces, linear dependence, dimension and bias. A few introductory problems proving certain sets to be vector spaces. ||Introductory problems related to Vector Spaces - Problems demonstrating the concepts introduced in the previous tutorial. Checking or proving something to be a sub-space, demonstrating that something is not a sub-space of something else, verifying linear independence; problems relating to dimension and basis; inverting matrices and echelon matrices.||More concepts related to Vector SpacesDefining and explaining the norm of a vector, inner product, Graham-Schmidt process, co-ordinate vectors, linear transformation and its kernel. Introductory problems related to these.||Problems related to linear transformation, linear maps and operators - Solved examples and problems related to linear transformation, linear maps and operators and other concepts discussed theoretically in the previous tutorial. |
|Definitions of Rank, Eigen Values, Eigen Vectors, Cayley Hamilton Theorem |
Eigenvalues, eigenvectors, Cayley Hamilton Theorem
|More Problems related to Simultaneous Equations; problems related to eigenvalues and eigenvectors Demonstrating the Crammer rule, using eigenvalue methods to solve vector space problems, verifying Cayley Hamilton Theorem, advanced problems related to systems of equations. Solving a system of differential equations . || A few closing problems in Linear AlgebraSolving a recurrence relation, some more of system of equations.|| |