Matrix Equations and Special Inverses

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. With numbers, x times five equals ten, and five times y equals ten. Are x and y the same? Of course! Both are two. Now the same thing with matrices: X times A equals B, and A times Y equals B, with the same A and B in both. Same answer again? Most students say yes, without thinking. Hmm, but is it really? Hold that thought. In Example two we will actually solve both, and you will see for yourself. Matrix equations come up often in JEE, so let's begin.

0:43Solving matrix equations: no dividing!

So, here is a matrix equation: A X equals B, with X the unknown matrix. Matrix A is invertible, so its determinant, in the straight bars, is not zero; the slashed equals sign means not equal. With numbers, five x equals ten, you just divide by five. So, can we divide B by A here? No, no! There is no division for matrices. Instead, we multiply by A inverse, the A with a small minus one on top. On which side? Matrix A sits on the left of X, so A inverse goes on the left, on both sides: A inverse A X equals A inverse B, with me? Now, A inverse times A is the identity matrix, capital I, and I times X is just X itself. So X equals A inverse B; the arrow means implies. Okay? Now flip it. Suppose X A equals B, so matrix A is on the right of X now. Is the answer again A inverse B? Oh no, careful! That is the trap. Matrix A is on the right, so we multiply on the right by A inverse. Then XAA inverse equals B A inverse, so X equals B A inverse. Matrix multiplication is not commutative, so B A inverse and A inverse B are usually different. Clear? And what if X is sandwiched, like AXC equals B? Put A inverse on the left and C inverse on the right: X equals A inverse B C inverse. Now both A and C must be invertible. Right? So, one simple rule: the inverse goes on the same side as the matrix it cancels. Left stays left, right stays right. Got it?

2:34Orthogonal matrices: the inverse for free

Now a special family: orthogonal matrices, from earlier in this unit. Matrix A, a square matrix, is orthogonal when A times A transpose, and A transpose times A, both give the identity matrix. The small T on top means transpose. Okay? So tell me: for an orthogonal matrix, what is A inverse? Yes, exactly! It is A transpose itself. So matrix A is orthogonal if and only if A inverse equals A transpose; the double-headed arrow means if and only if. Topper's trick: when you spot an orthogonal matrix, skip the cofactors and just transpose it. Quick, right? And the family stays together: products and inverses of orthogonal matrices are orthogonal too. With me? For example, the rotation matrix R of theta, theta the Greek letter, is orthogonal. So its inverse is just its transpose, and that is R of minus theta. Nice, isn't it? One more neat result. Take a real skew-symmetric matrix A, which means A transpose equals minus A, okay? Then I plus A is always invertible. Right? And I minus A, times the inverse of I plus A, is an orthogonal matrix. The proof that it is orthogonal is a solved example on this page. Clear?

3:59Similar matrices

Now the last family: similar matrices. If B equals P inverse A P, for some invertible matrix P, we say B is similar to A, okay? Careful: P inverse and P do not cancel here. They are not next to each other, and we cannot reorder a matrix product. Right? Now, what is B squared? Write P inverse A P twice, and look at the middle. Yes! In the middle, P times P inverse gives the identity, and it disappears. So B squared is P inverse, A squared, P, got it? Every extra factor cancels the same way, so B to the power n is P inverse, A to the power n, P, clear? Now the determinant. Determinants multiply, so determinant of B is determinant of P to the power minus one, times determinant of A, times determinant of P, okay? These are just numbers, so they cancel: determinant of B equals determinant of A, every time. And trace of B equals trace of A as well; t r is the trace, the sum of the diagonal entries. So similar matrices share their determinant and their trace. With me?

5:13Example 1: solve AX = B with a 2x2 matrix A

Alright, Example one, an easy one. In square brackets, the matrix with rows two, one and three, two multiplies an unknown matrix X, giving the column four, seven. So, what is X? Pause the video, and try it first. Done? Okay, let's check. Call the square matrix A, and the column on the right B, okay? Determinant of A: four minus three, one. Not zero, so the inverse exists. Now swap the two diagonal entries, change the signs of one and three, and divide by the determinant, which is just one: A inverse has rows two, minus one and minus three, two. Right? Now, X has matrix A on its left. So on which side does A inverse go? On the left, correct! So X equals A inverse B, okay? Top entry: two times four, minus one times seven, so eight minus seven. Bottom entry: minus three times four, plus two times seven, so minus twelve plus fourteen. With me? So X is the column with entries one and two. That's it! And always check: two times one plus one times two is four, and three times one plus two times two is seven. Perfect!

6:32Example 2: XA = B and AY = B give different answers

Now Example two, our puzzle from the start. Matrix A has rows two, one and five, three. Matrix B has rows one, zero and two, one. Solve X A equals B, and A Y equals B, with the same A and B in both. Pause and try. Hint: which side does A inverse go on? Done? Okay, let's check. Determinant of A: six minus five, one. Swap two and three on the diagonal, change the signs of one and five, and divide by the determinant, one again: A inverse has rows three, minus one and minus five, two. Okay? First, X A equals B: matrix A is on the right, so X equals B A inverse. Row one of B, one, zero, just picks out three, minus one. Row two is two, one: six minus five is one, and minus two plus two is zero. So X has rows three, minus one and one, zero. Right? Now, A Y equals B: matrix A is on the left, so Y equals A inverse B this time. Row one of A inverse is three, minus one: three minus two is one, and zero minus one is minus one. Row two is minus five, two: minus five plus four is minus one, and zero plus two is two. So Y has rows one, minus one and minus one, two. Got it? Now compare. Same A, same B in both equations. Are X and Y the same matrix? No! Completely different, exactly as expected. Puzzle solved: with matrices, the side really matters. Write A inverse B for both, and you lose one full answer.

8:24Quick revision: side of the inverse, orthogonal, similar

Clear? So, quick revision. Matrix equations: put the inverse on the same side as the matrix it cancels. If A X equals B, then X equals A inverse B; if X A equals B, then X equals B A inverse. Orthogonal matrices: the inverse is just the transpose. Similar matrices: B to the n is P inverse, A to the n, P, and they share the determinant and the trace. Now go and try the practice problems on this page. All the best!