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Linear Algerba - Matrices Part II - A Tutorial with Examples, Problems and Solutions

Linear Algebra


                                                                                                                         Linear Algebra: Matrices Part II

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 - Matrices Part II - Outline of Contents:

 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. 

In this tutorial, we introduce a few solved problems to help you understand the concepts introduced in Part-I.

Here are some quick notes on the kind of problems explained in this tutorial :

Q: In an upper triangular matrix n×n, minimum number of zeros is ... ?

Q: Let be a non singular matrix, 1+ p + p2+……..+ pn=0 (0 denotes the null matrix) then p-1= ... ?

Q: Finding the characteristic equation of a given matrix

Q: When does the inverse of a diagonal matrix exist (if at all) ?

Q: A trust fund had Rs. 50000 that is to be invested into two types of bonds. The first bond pays 5% simple interest per year and the second bond pays 6% simple interest per year.Using matrix multiplication determine how to divide Rs. 50000 among the two types of bonds so as to obtain an annual total interest of Rs.2780.

Q: Computing the inverse of a given matrix using elementary row transformations or otherwise.

Q: Finding the rank of a given matrix.

Q: Finding the circumstances under which a matrix will be invertible.

Q: Under what conditions does the inverse of a diagonal exist?

Q: Find all the matrix which commute with a given matrix.

Q: Compute the inverse of a matrix.

Q: Using elementary row transformations find the inverse of the matrix.

Q: N-th power of a matrix

Q: Systems of recurrence relations

Q: Rank of a matrix

Q: Let A and B be matrices of order n. Prove that if (I-AB) is invertible, then (I-BA)  is also invertible and (I-BA)-1 = I + B(I-AB)-1A


1. Linear Algebra by Kenneth Hoffman and Ray Kunze 

2. Linear Algebra by K. R. Matthews, University Of Queensland 

3. Mathematics by Amit M Agarwal

4. Mathematics by M.L Khanna

Tutorial with Solved Problems :

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.