Part C: Concept Checks

Transcript

0:00Introduction

Hello everyone! Okay, one quick challenge before we start. I read a statement, you tell me: true or false? For square matrices of the same order: is the determinant of A plus B equal to the determinant of A, plus the determinant of B? Looks completely natural, right? Many students tick true at once. Hold on to your answer; we will settle it today. Look, statements like this, true or false, or assertion and reason, come up often in JEE, and one careless tick costs marks. So, let's begin.

0:37Part C: how to decide true or false

So, what is Part C? Each item is a statement, and you decide: true or false. But a guess is not enough, okay? To say true, you need a one line reason that works in every case: every matrix, every order. Now think. To prove a statement false, how many failing cases do you need? Just one! Yes, exactly. One counterexample, one single matrix where it fails, and the statement is false. And the assertion reason type? Three questions, every time. Is the Assertion true? Is the Reason true? And if both are true, does the Reason actually explain the Assertion? Clear?

1:18Number rules that break for matrices

Now, the first trap. With numbers, if a times b is zero, then a is zero or b is zero, right? So for matrices: if A B equals O, the zero matrix, then A or B must be the zero matrix. The double arrow on screen means implies. True for matrices too? No, no, careful, that is false! Take the matrix with all four entries one, inside the square brackets, times the matrix whose first row is one and minus one, and whose second row is minus one and one. Every row of the first, dotted with every column of the second, gives zero. So the product is the zero matrix, but neither matrix is zero! Got it? The transpose of the product A B, inside the round brackets, equals A transpose times B transpose. The small T up top means transpose. What do you say? False again! The transpose of a product reverses the order: B transpose times A transpose. This is the reversal law, okay? The same reversal happens for the inverse. When A and B are invertible, the inverse of A B is B inverse times A inverse. The minus one up top means inverse. Right? And for the adjoint, written a d j on screen: for square matrices of the same order, the adjoint of A B equals the adjoint of B, times the adjoint of A, in that order. So always check the order first. With me?

2:40Determinant tools for one-line proofs

Alright, now determinants, the tools for one line proofs. The two vertical bars mean determinant. For square matrices of the same order n, the determinant of A B equals the determinant of A, times the determinant of B, okay? And the determinant of A to the power m is the determinant of A, to the power m. Now, multiply the whole matrix by a number k. Is the determinant of k A just k times the determinant of A? Oh no, careful! Multiplying the matrix by k multiplies each of its n rows by k, and each row gives one factor of k. So the determinant of k A is k to the power n, times the determinant of A, and transposing does not change the determinant. Got it? Let's use these on one assertion reason item. Assertion: if A cubed is O, the zero matrix, then the determinant of A is zero. Reason: the determinant of A cubed equals the determinant of A, cubed. Apply our three questions. Take determinants of both sides: the determinant of A, cubed, equals the determinant of the zero matrix, which is zero. So the determinant of A is zero. Both are true, and the Reason is exactly what proves the Assertion, so the Reason explains. Right?

4:00Example 1: a skew-symmetric matrix of order 3 is singular

Okay, Example one, an easy one. Assertion: every skew-symmetric matrix of order three is singular. Reason: the determinant of A transpose equals the determinant of A, and the determinant of minus A is minus one, to the power three, times the determinant of A, okay? Pause the video, and try it first. Done? Let's check. Skew-symmetric means A transpose equals minus A, the negative of the matrix. And singular means the determinant is zero; that is what we must prove. Clear? Now the chain. The determinant of A equals the determinant of A transpose. Since A transpose is minus A, that equals the determinant of minus A, right? And minus A is minus one times A, so by our k to the power n rule, this is minus one cubed times the determinant of A: just minus the determinant of A, with me? So the determinant of A equals minus itself. Bring it across: two times the determinant of A is zero, so the determinant of A is zero. Singular! Both statements are true, and the Reason is exactly what we used: the Reason explains. Now a quick trap. Does the same proof work for a skew-symmetric matrix of order two? No! For order two, minus one squared is plus one, so the chain only says that the determinant equals itself. Nothing new. Only odd order forces the determinant to be zero; an even order skew-symmetric matrix can be invertible. Remember that, okay?

5:34Example 2: four true or false statements on 3x3 matrices

Now Example two, a little harder: four statements, all for three by three matrices. One: the inverse of an invertible symmetric matrix is symmetric. Two: the adjoint of k A is just k times the adjoint of A, nothing more. Three: the determinant of A plus B equals the determinant of A, plus the determinant of B, our challenge from the start! Four: if the determinant of A is zero, then A times the adjoint of A is the zero matrix. Pause the video, and try all four. Done? Okay, let's check. Statement one. Symmetric means the transpose equals the matrix itself. Take the transpose of A inverse. Transpose and inverse can swap places, so this is the inverse of A transpose, and since A is symmetric, A transpose is A, so this is just the inverse of A again. The inverse is symmetric. True! Statement two. It looks like the number k simply comes out, right? Is it really just k? False! The adjoint of k A equals k to the power n minus one, times the adjoint of A, where n is the order. For order three, that is k squared. Why? Every entry of the adjoint is a cofactor, a two by two determinant, and each one picks up k squared. Clear? Now statement three, our challenge. Test it with the simplest matrices: take A and B both equal to I, the identity matrix. What do you get on each side? Left side: A plus B is two I, and its determinant is two cubed, eight. Remember k to the power n! Right side: one plus one, just two. Eight is not two, so it is false! Ticked true at the start? Many students do. That was the trap. Last one, statement four. The key identity: A times the adjoint of A equals the determinant of A, times the identity matrix. If the determinant of A is zero, this is zero times the identity: the zero matrix. True! So: true, false, false, true. Got it?

7:45Quick revision: counterexamples, reversal law, |kA|, adj kA

So, quick revision. False needs one counterexample; true needs a reason that works in every case. For products, the order reverses, and A B equal to O does not force either matrix to be zero. And watch the order n: k to the power n for the determinant, k to the power n minus one for the adjoint. And the slashed equals sign means not equal: the determinant of a sum is not the sum of the determinants. Now try the fourteen problem unit test on this page. Answers only, so check yourself honestly. All the best!