The Adjoint

Transcript

0:00Introduction

Hello everyone! Okay, a small magic trick first. Take the matrix with rows two, three and one, four. I swap the diagonal entries, and change the signs of the other two. Then I multiply the old matrix by the new one. Out comes five on the diagonal, and zeros everywhere else: five times the identity matrix! Why five? And why the zeros? Today we find out. It is called the adjoint, the key to the inverse of a matrix, which comes up often in JEE, okay? So, let's begin.

0:35From cofactors to the adjoint

First, a quick recap. Take a square matrix A, with entries a i j in square brackets; the small i j below is a subscript, for row i, column j. Capital C i j is the cofactor of a i j: minus one, to the power i plus j, times the minor M i j, the determinant left when you delete row i and column j. With me? Replace every entry by its own cofactor, and you get the matrix of cofactors: C i j, in square brackets. Right? So, is this matrix of cofactors the adjoint? What do you think? Not yet! The adjoint is the transpose of the matrix of cofactors; the small capital T at the top means transpose. We write adj of A; some books say adjugate. And it is defined for a square matrix, okay? The round brackets i comma j mean row i, column j. Because of the transpose, the indices swap: the entry there is C j i, not C i j. Clear? For three by three, look at column one of adj A: C one one, C one two, C one three, the cofactors of row one! So each row's cofactors go down the matching column. And the most common slip? Forgetting the transpose. Don't do that!

1:56The 2x2 shortcut for the adjoint

Now the two by two case; the cross in the heading is read as by. Take rows a, b and c, d. Delete a row and a column, and the one entry left is the minor. Can you find the four cofactors? Mind the chessboard signs. For a, only d is left, with a plus sign: C one one is d. For b, only c is left, in a minus place: C one two is minus c. The same way, C two one is minus b, and C two two is just a, right? Transpose, and the adjoint has rows d, minus b and minus c, a. Compare it with matrix A: what happened to each entry? Yes, exactly! The diagonal entries a and d swap places, and b and c stay put but change sign. Oh no, careful: do not swap b and c as well. That is the trap. Clear?

2:51The fundamental identity

Now the most important result. Matrix A times adj A equals adj A times A, equals determinant of A, times capital I, the identity matrix: ones on the diagonal, zeros elsewhere. The straight bars mean determinant, okay? But why? Entry i comma j of the product is row i of A, times column j of adj A, right? And what sits in column j of adj A? The cofactors of row j, because of the transpose! So the entry is a i one times C j one, plus a i two times C j two, plus a i three times C j three. With me? If i equals j, the entries and the cofactors come from the same row. What is the sum then? Correct! It is the expansion along row i, so every diagonal entry is determinant of A, right? And if i is not equal to j? The slashed equals sign means not equal. One row's entries with another row's cofactors: from the last chapter, that sum is zero. So the product is determinant of A, times the identity matrix. Got it? Now a tricky one. If determinant of A is zero, the matrix is singular, and A times adj A is the zero matrix, capital O, okay? Then must A, or adj A, be the zero matrix? No, no, careful! Take this example from the course: rows one, two and two, four, with determinant four minus four, zero. Its adjoint has rows four, minus two and minus two, one, and their product is the zero matrix. Yet neither factor is zero! Remember this one.

4:34Example 1: adjoint of two 2x2 matrices by the shortcut

Alright, Example one, an easy one. Find the adjoint of the matrix with rows two, three and one, four, and of the matrix with rows four, seven and one, two. Pause the video, and try it first. Done? Okay, let's check. Suppose you write the first matrix's cofactors in place: rows four, minus one and minus three, two. Is that the adjoint? No, no, careful! That is the matrix of cofactors; the transpose is missing. Use the shortcut: swap two and four, and change the signs of three and one. The adjoint has rows four, minus three and minus one, two. That's it! Now, our magic trick. Call the first matrix A, and multiply it by its adjoint: five on the diagonal, zeros elsewhere, so five times I, the identity matrix. And determinant of A is eight minus three, five. So the five was never magic: it was the determinant all along! Second matrix: swap four and two, and change the signs of seven and one. The adjoint has rows two, minus seven and minus one, four. Done in seconds! Clear?

5:48Example 2: adjoint of a 3x3 matrix, check A adj A = 22I

Now Example two, a step harder. Matrix A has rows one, two, three; zero, four, five; and one, zero, six. Find adj A, and verify that A times adj A is determinant of A, times the identity matrix. Pause the video, and try it. Done? Okay, let's check, row one first. For C one one, delete row one and column one: four, five, zero, six is left, so twenty four. For C one two, the minor is zero minus five, in a minus place, so plus five. And C one three is minus four. Okay? The same way, row two gives minus twelve, three, two, and row three gives minus two, minus five, four. Right? Now the trap. Is the first row of adj A twenty four, five, minus four? No! Transpose, remember? Row one's cofactors go down column one, row two's down column two, and row three's down column three. So adj A has rows twenty four, minus twelve, minus two; five, three, minus five; and minus four, two, four. With me? Now verify. Row one of A times column one: twenty four, plus ten, minus twelve, which is twenty two. That is just the expansion of the determinant along row one! Clear? Row one times column two: minus twelve, plus six, plus six, which is zero! One row's entries with another row's cofactors. Got it? And so on: twenty two on the diagonal, zeros elsewhere. So A times adj A is twenty two times I, and determinant of A is twenty two. Verified!

7:37Quick revision: adj A, 2x2 shortcut, A adj A = |A| I

So, quick revision. The adjoint is the transpose of the matrix of cofactors: row i's cofactors go down column i. Two by two: swap the diagonal, and change the sign of the other two. And A times adj A, in either order, is determinant of A times the identity matrix, even when the determinant is zero. In the next lesson, this gives us the inverse. Now go and try the practice problems on this page. All the best!