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## Our Linear Algebra Tutorials: at a glanceLinear Algebra - Matrices Part I - A Tutorial with ExamplesLinear Algerba - Matrices Part II - Tutorial with Problems and SolutionsLinear Algebra - Determinants - A Tutorial with Problems and SolutionsLinear Algebra - Simultaneous Equations in Multiple Variables Basic Concepts In Linear Algebra and Vector Spaces - A Tutorial with Examples and Solved ProblemsLinear Algebra - Introductory Problems Related to Vector SpacesLinear Algebra - More about Vector SpacesLinear Algebra - Linear Transformations, Operators and MapsLinear Algebra - Eigenvalues, Eigenvector,Cayley Hamilton TheoremLinear Algebra - Problems on Simultaneous Equations, EigenvectorsLinear Algebra - A few closing problems in Recurrence Relations                                                                                                                                                           ------------xxxx------------

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!

 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: 2n-1

Q:

Q:

Q:

Q:

Q:

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: is also invertible and (I-BA)-1-1

References:

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

### 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 IIProblems 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 spaces Introductory problems related to Vector Spaces More concepts related to Vector Spaces Problems related to linear transformation, linear maps and operators Definitions of Rank, Eigen Values, Eigen Vectors, 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.