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Once you know how to multiply 2[br]matrices together, a natural

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question to ask is whether or[br]not one matrix can be divided by

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another matrix, and the answer[br]to that question is no, there is

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no such thing as matrix

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division. Nevertheless, there's[br]another operation that we can

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introduce which plays a very[br]similar role to division, and

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it's called finding the inverse[br]of a matrix. So in this video

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I'm going to explain what we[br]mean by the inverse of a matrix.

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To start with, let's just look[br]at these two matrices that I've

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written down here, and let's[br]find the product of them. We

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multiply these two together.

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4 * 1 is 4 added to 3 *[br]-- 1 is 4 -- 3 which is 1.

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4 * -- 3 is minus 12, three 4 +[br]12 -- 12 + 12 is 0.

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1 * 1 is 1 and 1 * -- 1 is minus[br]one 1 -- 1 is or is not.

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1 * -- 3 is minus three 1 * 4 is[br]four 4 -- 3 is 1.

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The point I'm trying to make is[br]that when we multiply these two

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matrices together, the answer[br]that we get is an identity

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matrix. That point is important.

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Suppose we do the[br]multiplication in the opposite

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order, so 1 -- 3 -- 1 four[br]multiplied by 4311. So I've

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changed the order of the[br]multiplication there.

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One's voice 4 -- 3 * 1 is minus[br]three. 4 -- 3 is 1.

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Once three is 3 -- 3 * 1 is[br]minus three, 3 -- 3 is nothing.

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Minus 1 * 4 is minus 44144 and[br]the result of adding them

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together is 0.

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Minus 1 * 3 is minus 3414, four[br]4 -- 3 is 1, so again the result

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of multiplying these two[br]together is an identity matrix.

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This leads us to the following[br]definition for the inverse of a

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matrix. If you've got one matrix[br]and you multiply it by another

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matrix, and the answer is an[br]identity matrix, then the second

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matrix is the inverse of the[br]first matrix and vice versa. The

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first matrix is the inverse of[br]the second matrix. Now suppose

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we have a matrix A and we[br]multiply it by its inverse and

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we'll denote the inverse by A to[br]the power minus one. Now that's

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not really a power. That does[br]not mean 1 / A.

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This is the notation that[br]will use for the inverse of

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the matrix A, so if you have[br]a matrix a multiplied by its

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inverse.

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The answer is an identity[br]matrix. That's what we mean

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by an inverse matrix. The[br]same works the other way

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around as well, so if you[br]started with an inverse

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matrix and multiplied it by[br]the original matrix, the

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answer two would be an[br]identity matrix, and that's

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the important definition of[br]an identity matrix.

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If we want to find an inverse of[br]the matrix, there's a simple

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formula that exists when the[br]matrix is a two by two matrix.

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So let's have a look at the

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formula Now. Suppose we've[br]got the matrix ABCD.

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The formula for the inverse[br]is as follows. The inverse,

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which as I say is denoted by[br]A to the minus one.

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Is given by 1 divided by.

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The number is 8 * D. Subtract B[br]* C rather strange formula, but

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you'll see how it develops in a[br]minute. 1 / a D minus BC

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multiplied by the matrix that[br]you get when you interchange the

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A and the D round. So the DB[br]comes up here and the A goes

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down there. And we changed the[br]sign but leave leave these

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elements in place. The BNC in[br]place, but change their sign to

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minus BN minus C, and this[br]formula that we have here is the

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formula. Then for the inverse of[br]the matrix A.

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I'm going to apply this[br]formula to try and find the

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inverse of a matrix that I'll[br]give you here. Now, supposing

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the Matrix is a is 3142.

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Now applying the formula, we'll[br]get that the inverse matrix is

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equal to 1 / 8 * D. That's 3 *[br]2, which is 6 -- B * C, which is

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1 * 4.

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Multiplied by a matrix which you[br]get when you interchange the A

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and the D. So the A and the D

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swap round. So swapping the[br]three in the two round will

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get two and three.

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And changing the sign of the[br]other two elements, but leaving

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them in the same place.

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So this is the inverse of matrix

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A. Let's just tidy it[br]up a little bit 1 / 6

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-- 4 is 1 / 2 or 1/2.

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And we could leave the answer[br]like that. Or we can multiply

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the factor of 1/2 inside,[br]multiplying every element inside

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by the half. But I'm going to[br]leave it like that for now.

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So this is the inverse of matrix[br]A. If we want to check whether

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it's whether it's the right[br]answer or not, all we have to do

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is multiply these two together[br]and the answer that we should

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get should be an identity[br]matrix. So let's just check that

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what I'll do is I'll workout A[br]to the minus 1 * A.

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So here's A to the minus[br]one we've just found.

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And I'm going to multiply it by[br]a, which was three 142.

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Let's leave the half[br]outside for now.

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To do the matrix multiplication,[br]here we've got 2, three or six.

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Added 2 -- 1 * 4 that's 6 -- 4[br]which is 2.

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2 ones or two.

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Add it to minus 1 * 2. That's 2[br]-- 2, which is nothing.

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Minus 4 * 3.

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Plus 3 * 4, which is minus 12 +[br]12, which is nothing and finally

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minus 4 * 1, which is minus four[br]added to 3, two six 6 -- 4 is 2.

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And finally, if we multiply each[br]element by the factor of 1/2

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outside, you'll see that we get[br]the identity matrix, so that's a

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verification that this matrix we[br]have found here is the inverse

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of the matrix we started with.

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For yourself, you could verify[br]that we get the same results by

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multiplying them the opposite[br]way around. Finding a time Zeta

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minus one, and you'll still find[br]that you get an identity matrix,

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and that's how you use the[br]formula to find the inverse of a

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two by two matrix.

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Now there are certain situations[br]where this won't work, and the

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reason why it won't work is[br]obtained by looking at this

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quantity in the formula here.

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If it transpires that a D -- B C

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is 0. Would be trying to divide[br]by zero here.

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And division by zero is never

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possible. So what this means is,[br]if ABC&D are such that a D -- B

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C is 0.

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The inverse of this matrix[br]will not exist.

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Now you may remember that the[br]quantity AD minus BC we've met

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before. It's called the[br]determinant of this matrix A.

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So whenever the determinant of[br]the matrix A is 0.

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The inverse won't exist.

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Equivalently, whenever A is a[br]singular matrix, the inverse

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won't exist. Let's just have a[br]quick look at an example of that

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sort, supposing we had a matrix[br]A which was 326 fourth.

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If we find a D minus BC for this[br]matrix, that's 3 * 4, which is

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12. Subtract 2 * 6 which is 1212[br]-- 12 is zero. We see that the

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determinant of a is 0.

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This is a singular matrix, and[br]so because it's a singular

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matrix, no inverse matrix will

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exist. So the point to remember[br]is that not all two by two

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matrices will have an inverse.[br]Those that don't have an inverse

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accord singular matrices. But[br]when a matrix does have an

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inverse, this formula will[br]enable you to find it.