Homogeneous Systems

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. I give you three equations in x, y and z, and every right side is zero. There is one solution you can spot without doing any calculation at all. Can you see it? Keep your answer in mind; we will check it in a minute. But the real question is this. Is that the only solution, or are there others? One determinant will tell us. Questions like this come up often in JEE, so let's begin.

0:32Homogeneous: every right side is zero

So, here is our system: three equations in three unknowns, x, y and z, and every right side is zero. Such a system is called homogeneous. The letters a, b and c are the coefficients, and the small numbers at the bottom right are subscripts: a one, b one, c one belong to the first equation, and so on. Okay? In matrix form, A is the matrix of coefficients, and capital X is the column of unknowns. The right side is capital O, the zero matrix, a column of zeros. So the whole system is just one line: A X equals O, right? Now, remember our puzzle? Which solution can you see straight away, with no work at all? Yes, exactly! x, y and z all zero, that is, capital X equals the zero matrix. Zero times anything is zero, so every equation holds. We call this the trivial solution; trivial simply means obvious. So a homogeneous system always has at least one solution: it is always consistent. Clear?

1:36Only the trivial solution, or infinitely many?

Now the real question: is the trivial solution the only one? Delta, the Greek capital letter that looks like a little triangle, is the determinant of A, written with straight bars. Three equations in three unknowns, so A is square, three by three, and it has a determinant. Got it? Think back to Cramer's rule. The slashed equals sign means not equal, okay? So, when delta is not zero, how many solutions does a system have? And here, which one must it be? Exactly one solution! Correct. And we already know one solution here, the trivial one. So when delta is not zero, the trivial solution is the only one. Nothing else, right? Now, what if delta is zero? For a general system, A X equals B, delta zero can even mean no solution at all, right? So here also, maybe no solution? No, no, careful! That is the trap. Here the trivial solution is always there, so a homogeneous system can never have no solution. So when delta is zero, there are infinitely many solutions, including non-trivial ones, where x, y and z are not all zero. Clear? So, put the two cases together: non-trivial solutions exist exactly when delta is zero. So when a JEE question says non-trivial solution, your first move is simple: set delta equal to zero. With me?

3:02Topper's trick: the cross product of two rows

Alright, delta is zero. But how do we actually find those solutions? Here is a topper's trick. Take two rows of coefficients, written in round brackets. One condition: the two rows must not be proportional, which means one row is not simply a multiple of the other. Okay? Then, x is to y is to z; the colons mean ratio. It equals: b one c two minus b two c one, is to c one a two minus c two a one, is to a one b two minus a two b one. This triple is called the cross product of the two rows. Clear? Now look at the middle part. Many students write it as a one c two minus a two c one. Same thing, right? No, no, careful! That has the opposite sign. Here is how to remember it. The first part leaves out the a's, the second leaves out the b's, and the third leaves out the c's. And the first letters follow the cycle b, c, a, okay? So the middle part starts with c one, then a two. Got it? And when delta is zero, every solution is t times this triple, where t is any number. So delta zero tells you that solutions exist, and the cross product tells you what they are. With me?

4:15Example 1: does this system have a non-trivial solution

Alright, Example one, a warm up. Does x plus y plus z equals zero, x minus y plus z equals zero, and x plus two y minus z equals zero, have a non-trivial solution? Pause the video, and try it first. Done? Okay, let's check. The system is homogeneous, so delta decides everything. The coefficients go in row by row, between the straight bars: one, one, one; then one, minus one, one; then one, two, minus one. Right? Expand along the first row. First term: one, times, one minus two. Second term: minus one, times, minus one minus one. Third term: plus one, times, two plus one. Clear? So delta is minus one, plus two, plus three: four. Not zero! Okay? Now you tell me. Delta is four, not zero. So, is there a non-trivial solution? No! Correct. When delta is not zero, the trivial solution is the only one: x, y and z are all zero. So here, zero really is the only solution: our puzzle has its first answer. Careful: zero right sides alone do not guarantee infinitely many. Only delta zero gives that. Right?

5:36Example 2: find k and the ratio x : y : z

Now a harder one, JEE style. x plus y minus z equals zero, three x minus y minus z equals zero, and x minus three y plus k z equals zero. This system has a non-trivial solution. Find k, and find x is to y is to z. Pause the video, and try it. Done? Okay, let's check. The question says the system has a non-trivial solution. So what is our very first move? Yes, exactly! Set delta equal to zero. Row one: one, one, minus one. Row two: three, minus one, minus one. Row three: one, minus three, k. Okay? Expand along the first row. First term: one, times, minus k minus three. Second term: minus one, times, three k plus one. Third term: the entry is minus one, times, minus nine plus one. Collect: minus k minus three, minus three k minus one, plus eight. Total: four minus four k. Right? Set it to zero: four minus four k equals zero. The double arrow means gives: k equals one. Got it? Now the ratio. Take the first two rows, one, one, minus one, and three, minus one, minus one. Clearly not proportional, so our trick works. First part, leave out the a's: minus one minus one, that is minus two. Middle part, starting with c one, then a two: minus three plus one, that is minus two. Last part: minus one minus three, that is minus four. Clear? Divide every part by minus two: x is to y is to z equals one is to one is to two. With me? But wait. We used only the first two equations. What about the third one? Check it, like a topper! Third equation, with k equal to one: x minus three y plus z. Put in one, one, two: one minus three plus two, zero. Perfect! And remember our puzzle? Here, zero is not the only solution: there are infinitely many, all multiples of one, one, two. Flip the middle sign, and this check fails. That is why toppers always check!

8:01Quick revision: trivial or non-trivial, Delta = 0, x:y:z

So, quick revision. One: a homogeneous system, A X equals O, is always consistent, because the trivial solution always works. Two: delta not zero, only the trivial solution; delta zero, infinitely many. Three: for non-trivial solutions, set delta equal to zero, then take the cross product of two rows that are not proportional. Now go and try the practice problems on this page. You are ready for them now. All the best!