Overview

Transcript

0:00Introduction

Hello everyone! Okay, a quick challenge before we start. Take the two by two matrix with every entry equal to one. Raise it to the power ten, and add up all four entries of the answer. How long will that take you? Nine matrix multiplications by hand? That eats up your time, right? In this chapter, a question like this should take one to two minutes, no more. Matrices and determinants come up often in JEE Main, so this speed really matters. Today, I will show you how this chapter of mixed problems works, and how to use it well. Let's go.

0:38Why this chapter

So, first, why this chapter matters. Matrices and determinants give two to three questions in many JEE Main shifts. The little dash on screen is read as to, so two to three, okay? So this unit is really worth your time. Now, the real JEE Main questions, from twenty twenty one to twenty twenty six, are already at the end of Chapters one to four. You meet them there, chapter by chapter. Right? And this chapter is different. Its problems are original, written in the same style as those real ones. So everything here is fresh practice, questions you have not seen before. Got it?

1:20Five question types

Alright, now think about the four chapters of this unit. Which types of questions do you expect in the exam? Here they are, five types. One: powers and special matrices. Two: counting matrices, where you count how many matrices of a given kind there are. Okay? Three: nested adjoints, that is, an adjoint inside another adjoint. Four: properties of determinants. Clear? And five: systems of equations with parameters, where an unknown constant sits inside the equations. You will meet all five types in this chapter. With me?

1:58Three sets, one clock

Now, the problems come in three sets. Set one: matrix algebra and powers. Set two: determinants, adjoint and inverse. Right? And Set three: linear systems, that is, the systems of equations. Okay? Each set has fifteen solved problems, and each problem should take you one to two minutes. So tell me, how long should one whole set take? Fifteen to thirty minutes. Correct! Fifteen times one, and fifteen times two. So start a clock before you begin, and see where you stand. Clear? And the most important rule: attempt the whole set before reading any solution. Peeking at each answer feels comfortable. But it hides your real speed, and your real mistakes. So finish the set first, then check. With me?

2:50Example 1: sum of all entries of A^10, all-ones matrix

Now, remember our challenge from the start? Let's finish it. It is from Set one, and it is marked easy. Matrix A, inside these square brackets, has every entry equal to one. Find the sum of all the entries of A to the power ten; the small ten at the top right is the power, okay? Start your clock, pause the video, and try it. Done? Okay, let's check. We are not doing nine multiplications by hand. Here is a topper's trick: find A squared first, and look for a pattern. Row times column: one times one, plus one times one, gives two, in every place. So A squared is the matrix of all twos, and that is simply two times A, clear? Now, A squared is two times A, as we saw. So A to the power ten must be two to the power ten, times A, right? No, no, careful! That is the trap. Check A cubed first. The dot means times, so A cubed is A times A squared. That gives two A squared, which is four times A, okay? And four is two squared. So A cubed is two to the power two, times A, not two cubed. So, here is the pattern. Each extra A brings one more factor of two. From A up to A to the power ten, there are nine extra A's. So A to the power ten is two to the power nine, times A, got it? So every entry is two to the power nine, that is, five hundred and twelve. Four entries, so the sum is four times five hundred and twelve: two thousand and forty eight! About one minute, and done. That is the speed this chapter trains. Perfect!

4:35Example 2: |adj(adj(3A))| for a 3x3 matrix with |A| = 2

Now a medium one, from Set two: nested adjoints. Matrix A is three by three; the little cross is read as by. The two vertical bars mean determinant, so the determinant of A is two. And adj, with the round brackets, means the adjoint of whatever is inside. The determinant of adjoint of adjoint of three A is two to the power a, times three to the power b. Find a plus b. Pause the video, and try it. Done? Okay, let's check. First, a result from Chapter three. For a square matrix M of order n, the determinant of adjoint of M equals determinant of M, to the power n minus one. Here n is three, and we take the adjoint twice. So, what power of determinant of M do we get? Four. Yes, exactly! One adjoint gives determinant of M, squared. The second adjoint squares that again; the big round brackets hold the first square. Two times two is four, so we get determinant of M to the power four. With me? Now our M is three A, and we need its determinant. Easy: three times two, that is six, right? No, no, careful! This is the classic trap. In three A, every entry carries a factor of three, so one three comes out of each of the three rows. For a scalar k and a square matrix of order n, determinant of k A is k to the power n, times determinant of A, from Chapter two. So three cubed times two, that is fifty four, which is two times three cubed. Got it? So the determinant of adjoint of adjoint of three A is fifty four to the power four. Fifty four is two times three cubed. To the power four, that gives two to the power four, times three to the power twelve, since three times four is twelve. Clear? So a is four, b is twelve, and a plus b is sixteen! Look back at the third line: determinant of k A is k to the power n, times determinant of A, never just k times determinant of A, okay? Pull every scalar out first, and you are safe. That's it!

6:55Quick revision: three sets, powers and |kA| = k^n |A|

So, quick revision. Three sets of fifteen problems: one to two minutes each, and the whole set before any solution. For powers, find A squared first, then look for the pattern. And for scalars, determinant of k A is k to the power n, times determinant of A, never k times determinant of A, remember. Now open Set one, start your clock, and begin. You are ready for this, I promise. All the best!