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Every square matrix has[br]associated with it a special

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quantity called its determinant.

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And determinants turn out to be[br]very useful when it comes to

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doing more advanced things with[br]matrices like finding inverse

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matrices and solving[br]simultaneous equations and later

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videos will cover these topics[br]were going in this video. Just

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look at finding the determinant[br]of a two by two square matrix.

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So let's look at an example.

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Suppose we have a matrix A.

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Which is got entries[br]in it elements in it

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435 and minus one.

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So this is a two by two matrix.[br]It's a square matrix.

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Now the determinant is a single[br]value. It's a value that's

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generated by combining the[br]numbers within the matrix in a

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special way for two by two[br]matrices. The way isn't

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particularly hard, but equally[br]it's not particularly obvious

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that you would do it this way,[br]so we need to just see an

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example, so we want to find the[br]determinant of A and we normally

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denote that by the letters Det[br]being the first 3 letters of

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determinant. So we right.

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Debt of a.

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A moment working out[br]determinants instead of using

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round brackets around the[br]matrix, we use vertical lines.

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But the numbers within the[br]vertical lines are exactly

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the same.

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And now we have to combine[br]those numbers.

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The first thing we do is we look[br]at the numbers on the leading

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diagonal. That's the four and[br]the minus one. And what we do is

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we find their product, we

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multiply them together. So 4 *[br]-- 1 Now having dealt with the

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numbers on the leading diagonal,[br]we've got 2 numbers left. The

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numbers that aren't on the[br]leading diagonal, and so we

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multiply those two numbers[br]together. So that's 3 * 5.

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So we've got the product of the[br]numbers on the leading

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diagonal, and we've got the[br]product. The numbers that are

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not on the diagonal, and then[br]we just take this second

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product away from the first[br]product.

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And then we work it out 4 * -- 1[br]is minus four. 3 * 5 is 15, so

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we take away 15 -- 4 takeaway.[br]15 is minus 19.

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And so that's how we do every[br]single determinant for a two by

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two matrix. We simply find the[br]product of the numbers on the

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leading diagonal and take away[br]from that the product of the

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other two numbers. So let's do

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another example. Here's another[br]two by two matrix B with

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elements 6, two.

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Three and five.

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And to workout the[br]determinant of B.

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We find the product of the[br]numbers on the leading diagonal

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is 6 and five. Then we take away[br]the product of the two numbers.

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Those two numbers are two and

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three. So we're taking[br]away 2 * 3.

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So 6 * 5 is 32 *[br]3 is 630 takeaway 6, and

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we're getting 24.

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The one more example here. So we[br]got the matrix D.

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Which has got entries 64.

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Three and two. Now when we[br]workout, the determinant of D.

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New the product, the numbers on[br]the leading diagonal is 6 * 2.

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Then we take away the product of[br]the two numbers, which is 3 * 4.

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So 6 * 2 is 12. Taking away 3[br]* 4, she's also 12, and so we

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get the answer 0.

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So we see that every time we[br]workout the determinant, what we

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get is a single number.

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Determinant of a is minus 19 the[br]determinant of B is 24

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determinant of D is 0.

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Now when a matrix has a zero[br]determinant, it has a especially

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given name to that property. We[br]say that the matrix is singular.

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So any matrix which is singular[br]is a square matrix whose

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determinant is 0.

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These other matrices, where[br]the determinant wasn't zero

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are called nonsingular.

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So it doesn't matter what the[br]determinant is, just so long as

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it isn't zero. That makes the

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matrix nonsingular. Finally,[br]we'll just look at a general two

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by two matrix. So here we have a[br]matrix A and the entries the

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elements in this matrix RABC&D.

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If you want to workout[br]the determinant of a.

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We just follow the process we've[br]already seen. We take the values

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that are on the leading[br]diagonal. We multiply them

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together to get a D. We take the[br]other two values and multiply

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them together. It should be in C[br]and we subtract this second

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product away from the first[br]product, and so this is the

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general formula for any two by[br]two matrix. The elements are

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ABCD. The determinant is worked[br]out to be a * D minus.

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B * C.

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So that's all there is to know[br]about 2, but the determinants of

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two by two matrices.