Properties of the Adjoint

Transcript

0:00Introduction

Hello everyone! Okay, a small challenge first. Matrix A is three by three, with determinant two, and that is all I tell you. Find the determinant of the adjoint of three times the adjoint of two A, one inside the other. Sounds like a monster, right? By the end of this lesson, it will take you under a minute. And in JEE, these properties come up often. So, let's begin.

0:28Determinant of the adjoint

So, recall the big identity. Matrix A is square, n by n; the little cross is read as by. Matrix A times adjoint of A equals determinant of A, times capital I, the identity matrix. The straight bars mean determinant, and a d j is short for adjoint. Okay? Now take determinants on both sides. The right side is a number times the identity matrix. So, if k is a number, what is the determinant of k times the n by n identity? Many students write just k. No, no, careful! The identity is n by n, so the k sits in all n rows, and it comes out n times: k to the power n. The small n at the top right is the power. Got it? On the left, determinant of a product is the product of determinants: determinant of A, times determinant of adjoint of A, and on the right, determinant of A to the power n. Right? Divide by determinant of A, taken as not zero; the slashed equals sign means not equal. Property one: determinant of adjoint of A is determinant of A, to the power n minus one. For three by three, determinant of A, squared. Clear?

1:45The adjoint of the adjoint

Now, the adjoint of the adjoint. Use the big identity on adjoint of A itself. Adjoint of A, times adjoint of adjoint of A, equals determinant of adjoint of A, which is determinant of A to the power n minus one, times the identity matrix. With me? Now multiply both sides by matrix A, from the left. On the left side, matrix A meets adjoint of A first. What do these two give? Yes, exactly! Determinant of A, times the identity matrix. So the left side is determinant of A, times adjoint of adjoint of A, and the right side is determinant of A to the power n minus one, times matrix A, okay? Divide by determinant of A, which is not zero. Property two: adjoint of adjoint of A is determinant of A, to the power n minus two, times matrix A, right? Adjoint twice, and you are back to A itself, with a number in front! Now its determinant. Hint: adjoint of adjoint of A is the adjoint of one matrix, so use property one on it. What do you get? Very good! Determinant of adjoint of A, to the power n minus one. So the power is n minus one, the whole squared. Each adjoint multiplies the power by n minus one. So if the adjoint is taken r times, the determinant is determinant of A, to the power n minus one, raised to r. Got it? Now put n equal to three. Adjoint of adjoint of A is determinant of A, times matrix A, and its determinant is determinant of A, to the power four. Clear?

3:34Scalars, products, powers, transposes

Alright, now a number multiplies the matrix. Is the adjoint of two A just two times the adjoint of A? What do you think? No, no, careful! Take any number k. Each cofactor is a determinant of size n minus one, and each of its rows carries a k. So adjoint of k A is k to the power n minus one, times adjoint of A, okay? For three by three, that is k squared: so adjoint of two A is four times adjoint of A, not two times! Right? Next, the product of two n by n matrices. Adjoint of A B is adjoint of B, times adjoint of A: the order reverses, just like the reversal law for inverses, which proves it when A and B are invertible. With me? Powers and transposes are friendly. Adjoint of A to the power m is adjoint of A, to the power m, and adjoint of A transpose, the capital T at the top, is the transpose of adjoint of A, okay? Last, the inverse. Here determinant of A must not be zero; the minus one at the top means inverse. Adjoint of A inverse equals the inverse of adjoint of A, and both equal matrix A over its determinant. Clear?

4:54Topper's trick: work from the inside out

Now a topper's trick for nested expressions: work from the inside out, and keep every scalar in front. Try the one on the badge: the adjoint of two times adjoint of A, for three by three. What number comes out in front? Four! Yes, two squared. The two is inside an adjoint, so it comes out with power n minus one, two here. So: four times adjoint of adjoint of A, okay? And adjoint of adjoint of A is determinant of A, times matrix A, so the answer is four times determinant of A, times matrix A, in just two lines! Simple.

5:36Example 1: |adj A|, |adj(adj A)|, |A inverse| when |A| = 3

Example one, an easy one. Matrix A is three by three, with determinant three. Find the determinants of adjoint of A, of adjoint of adjoint of A, of A inverse, and of adjoint of A inverse. Pause the video, and try it first. Done? Okay, let's check. Here n is three, so determinant of adjoint of A is three squared, nine. Easy, right? Next comes adjoint of adjoint of A: for three by three, it is determinant of A times matrix A, so it equals three times matrix A, okay? So is its determinant three times three, nine? No, no, wait! In determinant of three A, the three comes out of all three rows: three cubed; the dot means times. Three cubed times three: eighty one. Or directly, three to the power four. Got it? Next, determinant of A inverse is one over determinant of A: one third. With me? Last, adjoint of A inverse: it is the adjoint of a three by three matrix, so its determinant is determinant of A inverse, squared: one ninth. So: nine, eighty one, one third and one ninth. Clear?

6:58Example 2: nested adjoint |adj(3 adj(2A))| when |A| = 2

Now, remember our monster? Let's finish it. Matrix A is three by three, with determinant two. Find the determinant of the adjoint of three times the adjoint of two A, inside out. Pause the video, and try it. Done? Okay, let's check. Innermost first: adjoint of two A is four times adjoint of A, since the two comes out squared. The three in front makes it twelve times adjoint of A, got it? Next, we need the determinant of twelve times adjoint of A, okay? Determinant of adjoint of A is two squared, four. So is it twelve times four, forty eight? No, no, careful! Twelve comes out of all three rows: twelve cubed, one thousand seven hundred and twenty eight. The cross means times. Times four: six thousand nine hundred and twelve. Right? Now the outer adjoint. For three by three, its determinant is the determinant inside, squared: six thousand nine hundred and twelve, squared. With me? In prime powers, six thousand nine hundred and twelve is two to the power eight, times three cubed. Square it: two to the power sixteen, times three to the power six. Our monster, done in under a minute, without a single entry! Great job.

8:22Quick revision: |adj A|, adj(adj A), adj(kA), adj(AB)

So, quick revision. Determinant of adjoint of A is determinant of A to the power n minus one; adjoint of adjoint of A is determinant of A to the power n minus two, times matrix A, adjoint of k A is k to the power n minus one, times adjoint of A, and adjoint of a product reverses the order. Nested ones: inside out, scalars in front. Now go and try the practice problems on this page. All the best!