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Linear Algebra - Determinants - A Tutorial with Examples, Problems and Solutions


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                                                                                                                        Linear Algebra: Determinants


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 - Determinants - Outline of Contents:

 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. 

Here's a quick outline of the topics covered in this tutorial

1. Determinants : The basics of computing a determinant. Explaining what elements are. Elements at the same horizontal level form a row, those in the same vertical line form a column, And introduction to the first order determinant, second order determinant, third order determinant. 
2. Computing the minors and co-factors of a matrix. A minor Mij is the value of determinant of matrix A with ith row and the jth column removed. The co-factor = (-1)(i+j) * Mij for the element aij
3. Expansion of Determinant: A determinant can be evaluated by taking elements of any row
 
or column and multiplying with their cofactors; and adding the results. 

Properties of Determinants :

1. If rows and columns are interchanged, the determinant remains unaltered.
2. If any 2 rows or columns of a determinant are interchanged, then the resulting 
determinant is the negative of the original determinant
3. If the elements of any row (column) are multiplied by a non-zero scalar k, then the 
determinant is multiplied by k.
4. If two rows(or columns) in a determinant have corresponding entries that are equal, 
the value of determinant is equal to zero.
5. If each entry of a determinant is written as sum of two or more constituents then the determinant can be expressed as a sum of two or more determinants.
6. If to each element of a line (row or columns) of a determinant be added the equi-multiple of the corresponding elements of one or parallel lines, the determinant remains unaltered.
7. If each entry in any row or a column is zero then value of a determinant is zero. 

Introducing the method to solve systems of linear equations using the determinant method/Cramer Rule

We introduce the idea of solving simultaneous equations using determinants.
Consider a system of m linear equations in n unknowns. We first learn how to express these as a product of matrices. 
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.


Tutorial with solved problems :
These problems will help you understand how to apply row and column transformations to compute determinants. Computing and simplifying determinants involving terms with factorials, log, trigonometric ratios, algebraic expressions and so on. 


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.