Class 12 Maths-Strategy to learn Matrices and Determinants
Once upon a time, there were two friends Sarah and John, who were enthusiastic about Mathematics. They were both in Grade 12 and their Math teacher Mr. Smith had just introduced Matrices and Determinants. One sunny afternoon, after school, Sarah and John decided to explore the world of matrices further.
They had learnt that matrices were arrays of numbers and determinants were special numbers associated with these matrices.They started with finding the determinant of a 2 by 2 matrix. Moving on to a 3 by 3 matrix, they decided to expand it by the first row, in order to find the determinant.
Sarah and John went on further to calculate determinants of larger matrices. Their passion for the subject grew and they realised that with perseverance and hard work they could solve any problem they set their mind to. The two friends graduated from high school and carried their liking for mathematics in their respective careers.
Matrices have a wide range of applications starting with Mathematics and moving on to Statistics, Economics, Computer Science, Physics and many more. Today's topic helps get you started with basic matrix terminology and its applications.
The Easiest topic you'll ever learn-
My students just love this topic because it's so incredibly easy. Even students who tend to have difficulty in Mathematics usually get matrices right. So, an important tip for you, is, you can score in Matrices.
A matrix is basically an arrangement of rows and columns. The horizontal entries are called rows and the vertical entries are called columns. We denote the entries in round brackets or square brackets.
A word of caution here, you have to use either round or square brackets only. I will be explaining important terms in matrices for you. These will help you in answering MCQ based questions.
Get your notebooks ready and let's start!
Difference between a rectangular matrix and a square matrix
In a rectangular matrix, number of rows is not equal to the number of columns. In a square matrix, number of rows is equal to the number of columns.
Types of matrices:Diagonal matrix, Scalar matrix and Identity matrix
A diagonal matrix is a square matrix in which all the non diagonal elements are zero. A scalar matrix is a diagonal matrix in which all the diagonal elements are equal.
An identity matrix is diagonal matrix in which all the diagonal entries are unity. A word of advice to you, remember these 3 definitions as a number of objective based questions are asked on this topic.
Zero or null matrix is a matrix in which all entries are zero. You can add or subtract two matrices of the same order. You can also perform scalar multiplication on matrices, that is, you can multiply each element by a scalar.
Moving on to the most important part, matrix multiplication. Note that two matrices A and B can be multiplied if the number of columns of the first matrix = the number of rows of the second matrix.
Matrix multiplication, Symmetric and Skew Symmetric matrices
Matrix multiplication need not be commutative but it is associative. Transpose of a matrix is the matrix obtained by interchanging the rows and columns . Again, you have a number of properties for transpose of a matrix.
In this context, you have symmetric and skew symmetric matrices. Every matrix can be expressed as the sum of a symmetric and skew symmetric matrix. This is a very important concept and you'll have a number of problems around this.
When is a matrix said to be invertible?
If A is a square matrix of order n and if there exists another square matrix of order n such that AB=BA=I, then B is called the inverse of A. If the inverse of A exists, A is said to be invertible.
Inverse of a matrix if it exists, is unique.
You need to learn the proof of this theorem as well, as it's mentioned in the syllabus. I've given you a gist of all the topics in matrices that you need to know. Podcast on formulas in matrices.
How I can help you learn Matrices in Mathematics!
I am an online tutor and I take classes one on one or in a group as per your requirement. You can join me for a webinar in Matrices for Class 12. The entire topic may take about four sessions to complete.
What's in it for you?
You'll learn definitions, concepts and most important, problem solving techniques. You'll also be able to solve MCQ based questions, assertion based questions and Case study based questions. These questions help prepare you for CUET exams as well.
You can improve your grades! You can email me on firstname.lastname@example.org for all queries pertaining to online tutoring. Looking forward to helping you.
Here are a few free resources for you!
Would you care to help other students of Class 12, who need that extra help. If yes, then please forward this blog to at least 5 students. Thank you!
Determinants and Matrices is one of the easiest topics in Class 12. Almost all of my students get this topic right. Just ensure that you have a knowledge of Matrices before you start learning Determinants.
What is a Determinant of a Matrix?
To start with, a determinant is a number attached to a square matrix. You will learn how to expand a determinant by any row or column. If A is a square matrix and k is any scalar, what is determinant of kA.
You can access all the formulas in the youtube link provided here.
You can use determinants to find the area of a triangle. This is an important concept. Note that, if the area of the triangle is zero, the points are collinear.
Minors and Cofactors
Minor of any element in a determinant of order n is a determinant of order n-1. Access the formula for minors and cofactors in this video. Once you learn how to calculate the cofactors of a matrix, you can progress to calculating the adjoint of a matrix.
Access more properties of adjoint of a matrix in the video link. A square matrix is said to be singular, if its determinant is equal to zero. And again, if a matrix is non singular its, inverse exists.
There are umpteen problems on this concept. Again, note that this is an important topic which carries considerable weightage in your exams.
Do you need extra help? There are a number of free resources below in the form of Youtube videos and multiple choice questions. I encourage you to make use of these.
Though this topic of Determinants is incredibly easy, you may still need some extra help in studying this. Feel free to register for my webinar on Determinants. The entire topic may require 4 to 5 classes of one hour each.
I will be teaching MCQ based questions, questions based on logical reasoning and case study based questions as well together with long answer questions. These are in tune with the NEP model as framed by ISC, NCERT. For further queries, you can contact me on email@example.com
I hope this was useful to you. You can access the formulas given in the links. Would you like to help other students who need that little extra help in Mathematics.
Until we meet again!