Products, Powers and Special Matrices

Transcript

0:00Introduction

Hello everyone! Okay, a quick puzzle first. You know that matrix multiplication is not commutative: matrix A times matrix B is usually not equal to B times A, right? So here is my question. For two square matrices of the same order, is the determinant of A B equal to the determinant of B A? Most students say no: the two products are different matrices, so how can they match? Hold that thought. In a few minutes, you will settle this yourself, in one line. And these rules come up often in JEE, in short questions with little traps. So, let's begin.

0:37Determinant of a product

So, here is the main result. Let A and B be square matrices of the same order, n by n. The vertical bars, as you know, mean determinant. Then determinant of A B equals determinant of A, times determinant of B, okay? The determinant of a product is the product of the determinants. Now, our puzzle. Determinant of A and determinant of B are just numbers, and numbers can be multiplied in any order. So what can you say about the determinant of B A? Yes, exactly! Determinant of A times determinant of B is the same as determinant of B times determinant of A, and that is determinant of B A, right? The slashed equals sign means not equal: so this holds even when A B is not equal to B A, okay? Different matrices, same determinant. Puzzle solved! Next, powers. The small m at the top is a power: A multiplied by itself, m times. So the determinant of A to the power m, with the power inside the bars, equals determinant of A raised to the power m, with the power outside. Simple, right? Two results you already know go with these. For an n by n matrix, determinant of k A equals k to the power n, times determinant of A, okay? And the small capital T means transpose: determinant of A transpose is the same as determinant of A, got it?

2:04Special matrices and their determinants

Now let's use these rules on the special matrices you met earlier. First, orthogonal: A times A transpose is the identity matrix, capital I, okay? Take the determinant of both sides. What can determinant of A be? Plus or minus one, correct! The left side gives determinant of A times determinant of A transpose, the square of determinant of A, and the identity matrix has determinant one. So determinant of A is plus or minus one: that is the plus-minus sign on screen. Clear? Involutory means A squared equals I, the identity. Same story: the square of determinant of A is one, so again plus or minus one. Right? Now idempotent: A squared equals A itself. Take determinants again. What are the possible values? The square of determinant of A equals determinant of A, so determinant of A is zero or one. Oh no, careful here! Do not cancel determinant of A from both sides, because it could be zero. That is how students lose the answer zero. With me? Nilpotent: some power of A, say A to the power k, is the zero matrix, capital O, okay? Then determinant of A, to the power k, is zero, so determinant of A is zero. Simple! Last one, and it is a classic. Skew-symmetric means A transpose equals minus A, remember? Take a skew-symmetric matrix of order three. What is its determinant? Zero, always! Watch. Determinant of A equals determinant of A transpose, which is determinant of minus A, right? Minus A is minus one times A, so by the k to the power n rule, we get minus one, in brackets, to the power n, times determinant of A, okay? For odd n, that is minus determinant of A, so determinant of A equals its own negative: zero! But remember the condition: odd order.

4:07Topper's trick: multiply determinants row by row

Now a topper's trick. Since determinant of B transpose equals determinant of B, determinant of A times determinant of B is also determinant of A times determinant of B transpose. And that is the determinant of A B transpose, okay? Why is that useful? The columns of B transpose are the rows of B, right? So the entry i comma j, in brackets, is row i of A dot row j of B: multiply matching entries, and add. Two determinants can be multiplied row by row. Got it? Let's see it work on a course example. Take M, the two by two matrix in square brackets, with rows a, b and c, d. Multiply M by M transpose, row by row. What do you get? Row one dot row one: a squared plus b squared. Row one dot row two: a c plus b d, and row two dot row one gives the same. Row two dot row two: c squared plus d squared. Right? So this determinant, in straight bars, is the determinant of M times M transpose: the square of determinant of M, which is a d minus b c, the whole squared. No expansion needed! When a determinant should be a square, think of this trick.

5:21Example 1: determinants of 2AB^T, A^2 B^3 and -A

Clear? Alright, Example one, an easy one. Matrices A and B are three by three, with determinant of A equal to two, and determinant of B equal to minus three. Find the determinant of two A B transpose, of A squared B cubed, and of minus A, okay? Pause the video, and try it first. Done? Okay, let's check. First one: two, times determinant of A, times determinant of B transpose. Two times two times minus three, so minus twelve, right? No, no, careful! The two multiplies the whole three by three matrix, so it comes out as two cubed, eight. And determinant of B transpose is just minus three. The dot means times: eight times two times minus three is minus forty eight. That's it! Second: split the product, then use the power rule: two squared, times minus three cubed. Four times minus twenty seven: minus one hundred and eight. Watch the sign: an odd power of a negative number stays negative. Okay? Third: minus A is minus one times A, so by the k to the power n rule, minus one cubed, times two: minus two. For a three by three matrix, the minus sign flips the sign of the determinant. Got it?

6:44Example 2: possible values of |A| when A^T A = 4I

Now Example two, a step harder. Matrix A is three by three, and A transpose times A equals four I, four times the identity matrix. Find the possible values of determinant of A, okay? Pause and try. Hint: it is just like the orthogonal case. Done? Okay, let's check. Take determinants of both sides. The left side is determinant of A transpose times determinant of A, so the square of determinant of A equals the determinant of four I, right? Now, what is the determinant of four I? The identity has determinant one, so four times one, just four? Oh no, careful! Four I is four times a three by three matrix, so the four comes out cubed. Four cubed, times the determinant of I, which is one: sixty four. That is the trap in this question. With me? So the square of determinant of A is sixty four, and determinant of A is plus eight or minus eight. Same idea as the orthogonal case. Nice and quick, right?

7:48Quick revision: |AB|, |A^m|, |kA| and special matrices

So, quick revision. The determinant of a product is the product of the determinants; a power can come outside the bars; k comes out as k to the power n; and transpose changes nothing. Orthogonal and involutory: plus or minus one. Idempotent: zero or one. Nilpotent, and odd skew-symmetric: zero. And to show a determinant is a square, multiply row by row. Now go and try the practice problems on this page. All the best!