1 00:00:01,050 --> 00:00:03,190 The next topic is Inverse Matrices. 2 00:00:07,320 --> 00:00:12,740 A square matrix A is invertible so square means the dimensions have to match, right? 3 00:00:12,740 --> 00:00:18,710 So, I've been calling matrices m by n, so m rows and n columns. 4 00:00:18,710 --> 00:00:24,240 Square just means that m is equal to n so I could say it's either m by m or n by n. 5 00:00:24,240 --> 00:00:26,350 And I think most of the time, I'll say n by n. 6 00:00:28,230 --> 00:00:32,790 so A is invertible if there exists a matrix, so I'm going 7 00:00:32,790 --> 00:00:38,484 to call the inverse A to the minus 1 power, it doesn't mean 1 over A. 8 00:00:38,484 --> 00:00:44,450 But it's sort of in the same sense that if I had a real number that's not equal to 0. 9 00:00:44,450 --> 00:00:47,375 And I took 1 divided by that real number, then 10 00:00:47,375 --> 00:00:51,260 when I multiply those two things together, I would get 1. 11 00:00:51,260 --> 00:00:54,150 So A inverse is going to be a matrix where I 12 00:00:54,150 --> 00:00:58,485 multiply either before or after A so I can take A-inverse A, 13 00:00:58,485 --> 00:01:01,715 that has to be equal to the identity matrix and 14 00:01:01,715 --> 00:01:05,470 A A-inverse has to be equal to the identity matrix. 15 00:01:07,540 --> 00:01:12,815 So if I can find this matrix A-inverse, then I will say that A is invertible. 16 00:01:18,450 --> 00:01:21,260 So the inverse, if it exists, is unique. 17 00:01:23,410 --> 00:01:29,106 And one way you can see that is if, so, so, what I'm trying to do here is I'm 18 00:01:29,106 --> 00:01:31,331 trying to say is that B and C are 19 00:01:31,331 --> 00:01:36,858 two different matrices that satisfy the properties in the definition. 20 00:01:36,858 --> 00:01:42,148 So the left-hand inverse, this one here, will be the 21 00:01:42,148 --> 00:01:47,950 matrix B and the right-hand inverse will be this matrix C. 22 00:01:47,950 --> 00:01:48,706 And then 23 00:01:48,706 --> 00:01:53,766 you can do a clever little trick. So 24 00:01:53,766 --> 00:01:58,726 if I have B times AC, but 25 00:01:58,726 --> 00:02:04,840 what is AC? That's just the identity matrix. 26 00:02:04,840 --> 00:02:09,284 So this is B times the identity matrix, which is just 27 00:02:09,284 --> 00:02:14,178 equal to B. But since I know AC, the product 28 00:02:14,178 --> 00:02:19,290 AC is also equal to the identity matrix. 29 00:02:19,290 --> 00:02:23,100 Then I can substitute the value AC in here for I. 30 00:02:24,390 --> 00:02:26,727 And then for my rules of matrix multiplication, 31 00:02:26,727 --> 00:02:29,110 it doesn't matter where I put the parenthesis. 32 00:02:29,110 --> 00:02:34,112 So I'm going to, instead of putting the parenthesis around AC, I'll put 33 00:02:34,112 --> 00:02:39,340 the parentheses around BA. But I've already assumed that this is 34 00:02:39,340 --> 00:02:42,880 equal to the identity matrix. 35 00:02:42,880 --> 00:02:47,150 So this is telling me that this is I times C which is the same thing as C. 36 00:02:49,570 --> 00:02:56,290 So if I assume BA is equal to the identity matrix and AC is equal to the identity 37 00:02:56,290 --> 00:03:03,360 matrix, then you can make an argument like this to say that B has to be equal to C. 38 00:03:03,360 --> 00:03:08,860 So there can only be one matrix that satisfies this property 39 00:03:08,860 --> 00:03:13,660 or these two properties if an inverse exists. 40 00:03:17,874 --> 00:03:25,219 So, if A is invertible, then the unique solution to this system of equations 41 00:03:25,219 --> 00:03:32,240 represented by Ax equals b, I'll start off just by writing Ax equals b. 42 00:03:32,240 --> 00:03:35,709 If A is invertible, then I have this matrix A inverse. 43 00:03:37,030 --> 00:03:41,814 So I'm, I'm going to multiply both sides of this, pre-multiply by A-inverse. 44 00:03:44,390 --> 00:03:49,276 So I put a A-inverse before the Ax, and an A-inverse before the b. 45 00:03:49,276 --> 00:03:51,901 And then A-inverse A, by the definition 46 00:03:51,901 --> 00:03:55,250 of the inverse, that's just the identity matrix. 47 00:03:55,250 --> 00:04:02,730 So identity matrix times x is equal to x. So I end up with x, which 48 00:04:02,730 --> 00:04:10,010 is the solution, this is the thing I'm looking for, is equal 49 00:04:10,010 --> 00:04:15,400 to A-inverse times b. And if there's a vector x 50 00:04:15,400 --> 00:04:20,800 that's not equal to 0 but the, has the property that 51 00:04:20,800 --> 00:04:25,850 A times x is equal to 0, then A is not invertible. 52 00:04:25,850 --> 00:04:30,303 And basically, I want to use this result here just to show what's going to go 53 00:04:30,303 --> 00:04:36,024 wrong so this, this is a kind of common argument that you make in mathematics. 54 00:04:36,024 --> 00:04:43,240 I want to assume that x is not equal to 0. 55 00:04:43,240 --> 00:04:46,975 And then based on that assumption, I want to prove that it is equal to 0. 56 00:04:48,100 --> 00:04:53,110 And so I'm, I'm assuming that something is true, contradicting that assumption. 57 00:04:53,110 --> 00:04:55,260 And so that's going to be called a proof by contradiction. 58 00:04:57,520 --> 00:05:00,190 And that's going to allow me to say that A is not invertible. 59 00:05:01,620 --> 00:05:05,662 Because if A is invertible, I can solve for this vector x 60 00:05:05,662 --> 00:05:11,140 here using this trick that I've just done in the third bullet point. 61 00:05:11,140 --> 00:05:11,575 [COUGH] 62 00:05:11,575 --> 00:05:14,968 So if there is a vector, a non-zero vector x so 63 00:05:14,968 --> 00:05:19,210 at least one element of this vector x has to be non-zero. 64 00:05:19,210 --> 00:05:23,690 Then I could write x as the identity matrix times x. 65 00:05:27,200 --> 00:05:29,696 And then if A was invertible, it would 66 00:05:29,696 --> 00:05:32,660 have an inverse and then that would satisfied 67 00:05:32,660 --> 00:05:35,312 the, the property in the definition so I 68 00:05:35,312 --> 00:05:39,039 could replace this identity matrix with A-inverse A. 69 00:05:43,910 --> 00:05:46,142 And then I can just, because of my rules for 70 00:05:46,142 --> 00:05:49,750 matrix multiplication, I can put parentheses wherever I want to. 71 00:05:49,750 --> 00:05:55,911 So I'm going to put some around this Ax here but I've assumed that there's 72 00:05:55,911 --> 00:06:01,540 this non-zero vector x that has a property that Ax is equal to 0. 73 00:06:01,540 --> 00:06:06,560 So x, remember, is non-zero. It has at least one non-zero component. 74 00:06:08,980 --> 00:06:12,390 But I can replace this Ax with the zero vector. 75 00:06:15,680 --> 00:06:20,430 And so that tells me that x has to be equal to A-inverse times 0. 76 00:06:20,430 --> 00:06:25,270 But anything times 0 is equal to 0. So what I've assumed on the 77 00:06:25,270 --> 00:06:30,550 left is that if A is invertible or I've assumed on the 78 00:06:30,550 --> 00:06:35,830 left that I have an invertible matrix a and this non-zero 79 00:06:35,830 --> 00:06:41,280 vector x. And then I've showed that A-inverse times, 80 00:06:41,280 --> 00:06:45,880 well, so essentially what I've been able to show with 81 00:06:45,880 --> 00:06:50,406 that assumption is that my non-zero x is equal to 0. 82 00:06:50,406 --> 00:06:52,996 So that's a contradiction so I can't have a 83 00:06:52,996 --> 00:06:56,200 non-zero vector x having every element equal to 0. 84 00:06:56,200 --> 00:07:01,650 And so then whatever I assumed has to be false. 85 00:07:01,650 --> 00:07:05,920 So I can't assume that A is invertible. 86 00:07:09,750 --> 00:07:11,580 Or if I assume that A is invertible, 87 00:07:11,580 --> 00:07:14,400 I can provide that it's not invertible essentially. 88 00:07:15,480 --> 00:07:19,560 So that's how you show that if there is this non-zero vector 89 00:07:19,560 --> 00:07:23,660 x, it satisfies Ax equals to 0, then A is not invertible. 90 00:07:27,890 --> 00:07:31,900 In the 2 x 2 case, there's actually a nice formula for computing the inverse. 91 00:07:39,220 --> 00:07:46,112 So, a 2 x 2 matrix will be invertible if and only if ad minus bc. 92 00:07:46,112 --> 00:07:47,840 So I've, I'm just labeling the elements. 93 00:07:47,840 --> 00:07:54,030 So I have a, b, c, d as the, the so ab is the first row, cd is the second row. 94 00:07:54,030 --> 00:08:00,260 And it needs to satisfy this property, so the ad minus bc has to be non-zero. 95 00:08:00,260 --> 00:08:04,300 If that's not equal to 0, then the inverse of A 96 00:08:04,300 --> 00:08:07,940 is going to equal to 1 divided by ad minus bc. 97 00:08:07,940 --> 00:08:12,971 So this is why this value has to be non-zero because 98 00:08:12,971 --> 00:08:18,410 I'm going to divide by it times d minus b minus ca. 99 00:08:18,410 --> 00:08:21,894 So that's going to give me a matrix that when I multiply it either 100 00:08:21,894 --> 00:08:25,480 in front of or after A, is going to give me the identity matrix. 101 00:08:27,940 --> 00:08:31,570 And the number ad minus bc is called the determinant of A. 102 00:08:31,570 --> 00:08:34,382 So essentially, if this number is not equal 103 00:08:34,382 --> 00:08:36,990 to 0, then A is going to be invertible. 104 00:08:44,240 --> 00:08:47,725 So a matrix is invertible if its determinant is not equal to 0. 105 00:08:50,940 --> 00:08:53,892 And then just, as a special case, I want to point 106 00:08:53,892 --> 00:08:57,426 out the easiest type of matrix to compute the inverse of. 107 00:08:57,426 --> 00:08:59,251 So the 2 x 2, I have this nice 108 00:08:59,251 --> 00:09:03,200 formula here, not nice necessarily, but I have a formula. 109 00:09:03,200 --> 00:09:09,536 A diagonal matrix just means that any off-diagonal element. 110 00:09:09,536 --> 00:09:12,371 So any, any element where I is not equal to 111 00:09:12,371 --> 00:09:16,097 J, so above the diagonal or below the diagonal, all of 112 00:09:16,097 --> 00:09:21,638 those have to be 0. If I have a matrix like that, 113 00:09:21,638 --> 00:09:27,110 then the inverse is just 1 over the diagonal elements. 114 00:09:27,110 --> 00:09:30,636 And so this also supposes that, so A-inverse will 115 00:09:30,636 --> 00:09:34,532 exist if all of these diagonal elements are non-zero. 116 00:09:34,532 --> 00:09:37,796 If if one of these, so suppose the first one was equal 117 00:09:37,796 --> 00:09:41,196 to 0, then there's nothing I could put in this cell here so 118 00:09:41,196 --> 00:09:44,010 that, that matrix wouldn't have an inverse. 119 00:09:46,450 --> 00:09:50,542 So if, if the diagonal elements of a diagonal matrix are all non-zero, then I 120 00:09:50,542 --> 00:09:52,836 can very easily calculate the inverses just 121 00:09:52,836 --> 00:09:55,610 by taking the reciprocal of each diagonal element. 122 00:09:55,610 --> 00:09:55,610 >> 123 00:09:55,610 --> 00:09:56,110 [INAUDIBLE] 124 00:09:59,240 --> 00:10:01,506 >> All of the rest of them stays 0. >> 125 00:10:01,506 --> 00:10:01,514 [INAUDIBLE] 126 00:10:01,514 --> 00:10:02,330 >> Yeah. 127 00:10:02,330 --> 00:10:06,330 And you can just convince yourself of this when you do the multiplication. 128 00:10:06,330 --> 00:10:13,023 I'll have first first row times the first column here so I'd have d1 times 1 over 129 00:10:13,023 --> 00:10:19,421 d1 and then just a whole bunch of zeros here times a whole bunch of zeros here. 130 00:10:19,421 --> 00:10:22,886 And if the I is not equal to J, then the non-zero element here will 131 00:10:22,886 --> 00:10:26,603 just crash into a 0 on this side and the non-zero element here will hit 132 00:10:26,603 --> 00:10:28,100 a 0 element on this side. 133 00:10:28,100 --> 00:10:30,239 So everything ends up being 0 off the 134 00:10:30,239 --> 00:10:32,810 diagonal so I get the identity matrix back. 135 00:10:36,640 --> 00:10:40,420 So then we have some rules for inverses. 136 00:10:40,420 --> 00:10:42,220 So these are probably the most important things 137 00:10:42,220 --> 00:10:44,561 you're going to want to remember about the inverse. 138 00:10:44,561 --> 00:10:47,045 generally, if you ever need to compute the 139 00:10:47,045 --> 00:10:49,529 inverse of a matrix, you're going to do that 140 00:10:49,529 --> 00:10:52,082 using a computer so even this formula here, 141 00:10:52,082 --> 00:10:55,060 it's not something I have committed to memory. 142 00:10:55,060 --> 00:10:57,780 If I need it, I'll look it up or it's usually, 143 00:10:57,780 --> 00:11:01,656 if you remember just the determinant, you can figure out what the, 144 00:11:01,656 --> 00:11:06,040 the arrangement of the rest of the letters has to be pretty quick, quickly. 145 00:11:06,040 --> 00:11:06,540 [COUGH] 146 00:11:08,410 --> 00:11:12,450 What is useful though is thinking about matrices sort of more abstractly. 147 00:11:12,450 --> 00:11:14,901 So instead of thinking about the actual elements, 148 00:11:14,901 --> 00:11:17,930 I'm trying to do mathematics with the matrices themselves. 149 00:11:19,510 --> 00:11:24,757 And so if I end up with a product AB and I need to take the inverse of 150 00:11:24,757 --> 00:11:30,260 that product, that ends up being B-inverse times A-inverse. 151 00:11:30,260 --> 00:11:33,554 And the way to convince yourself that's true is to just plug that into the 152 00:11:33,554 --> 00:11:34,920 definition of the inverse. 153 00:11:36,560 --> 00:11:40,385 So remember, my, my definition of the inverse said that if I 154 00:11:40,385 --> 00:11:44,840 have the inverse times the matrix, that has to give me the identity. 155 00:11:44,840 --> 00:11:49,240 So in this case, it would be AB inverse times the matrix product AB. 156 00:11:49,240 --> 00:11:52,123 And now this, I can write without the 157 00:11:52,123 --> 00:11:55,810 parentheses and that would be the same thing. 158 00:11:57,210 --> 00:11:58,659 The AB inversed, the order 159 00:11:58,659 --> 00:12:01,935 swapped, so it becomes B-inverse A-inverse and then I'll 160 00:12:01,935 --> 00:12:04,900 put some parentheses around what I get on the inside. 161 00:12:04,900 --> 00:12:08,426 A-inverse A is just equal to the identity matrix 162 00:12:08,426 --> 00:12:12,510 so it's going to just sort of collapse in upon itself. 163 00:12:12,510 --> 00:12:15,038 So this I, I can multiply that I times B 164 00:12:15,038 --> 00:12:20,000 that's identity matrix times anything just gives me the anything back. 165 00:12:20,000 --> 00:12:23,770 So I times B gives me B back. I end up with 166 00:12:23,770 --> 00:12:28,890 B-inverse B which is by the definition of an inverse, the identity matrix. 167 00:12:30,470 --> 00:12:35,090 And then I can do the same thing for the, for the right-hand inverse. 168 00:12:35,090 --> 00:12:39,170 Again, because it switches the order, all that's going to happen is I 169 00:12:39,170 --> 00:12:44,050 end up eliminating the B first so the B multiplied by B-inverse becomes identity. 170 00:12:44,050 --> 00:12:49,048 And I end up with AIA or AIA-inverse which becomes 171 00:12:49,048 --> 00:12:52,420 AA-inverse, which becomes I. 172 00:12:52,420 --> 00:12:56,848 And you can do exactly the same thing for a longer products as well. 173 00:12:56,848 --> 00:13:00,313 So, if I have ABC-inverse, I just reverse the order 174 00:13:00,313 --> 00:13:04,556 of the product so that becomes C-inverse B-inverse A inverse. 175 00:13:04,556 --> 00:13:09,138 So if you wanted to prove that that's true, you're just putting ABC 176 00:13:09,138 --> 00:13:14,115 over here and then its going to collapse three times rather than just two times 177 00:13:14,115 --> 00:13:17,670 but what you're left with is the identity matrix. 178 00:13:21,598 --> 00:13:25,405 So now, let's think about how would I actually find A-inverse. 179 00:13:28,250 --> 00:13:30,860 So I need to find a matrix so that when 180 00:13:30,860 --> 00:13:35,530 I multiply A by A-inverse, I get the identity matrix. 181 00:13:35,530 --> 00:13:38,962 If I can find this, I've already argued that it's going to be 182 00:13:38,962 --> 00:13:42,400 unique and it would also be the matrix that can go on the front. 183 00:13:42,400 --> 00:13:46,160 So, I don't need to compute both of them, I just need to compute one of them. 184 00:13:46,160 --> 00:13:49,756 If there's one matrix that have this property then it automatically 185 00:13:49,756 --> 00:13:53,414 has the property that if I pre-multiply by A-inverse, that's also 186 00:13:53,414 --> 00:13:55,520 going to turn A into the identity matrix. 187 00:13:59,410 --> 00:14:01,800 So, let's make some vectors here. 188 00:14:01,800 --> 00:14:05,619 So, I've used e before to mean a column vector of all ones. 189 00:14:07,860 --> 00:14:11,676 I'm going to add a subscript now to mean it's a vector of all zeros 190 00:14:11,676 --> 00:14:13,620 except it's got a 1 in whatever 191 00:14:13,620 --> 00:14:17,460 position, this little script, subscript corresponds to. 192 00:14:17,460 --> 00:14:20,190 So e sub 1 would be 1 0 0. 193 00:14:20,190 --> 00:14:27,005 and if it's in general in N dimensions, then I have e1 up through en. 194 00:14:27,005 --> 00:14:29,891 And it just means that it's all 0 except for 195 00:14:29,891 --> 00:14:33,167 the, the ith element and that's going to be equal 196 00:14:33,167 --> 00:14:33,644 to 1. 197 00:14:33,644 --> 00:14:37,340 So then, I have e2 is 0 1 0 and e3 is 0 0 1. 198 00:14:37,340 --> 00:14:40,304 And then, if I bind all three of these vectors 199 00:14:40,304 --> 00:14:45,210 together, e1, e2, e3, that's just going to be the identity matrix. 200 00:14:45,210 --> 00:14:48,880 So, you can almost kind of see it here. I just have ones going down the diagonal. 201 00:14:54,720 --> 00:14:57,216 And now I can imagine that I have an 202 00:14:57,216 --> 00:15:00,950 inverse, so A-inverse, an inverse of this matrix A. 203 00:15:01,950 --> 00:15:07,390 And the columns of my matrix A-inverse, I'm going to call x1, x2, and x3. 204 00:15:07,390 --> 00:15:10,740 So x1 is a column vector with three entries, x2 is a column 205 00:15:10,740 --> 00:15:15,010 vector with three entries, and x3 is a column vector with three entries. 206 00:15:16,420 --> 00:15:19,990 So I can write out my A A-inverse is equal 207 00:15:19,990 --> 00:15:26,860 to A times these three column vectors bound together into a matrix. 208 00:15:26,860 --> 00:15:30,859 And I know that has to equal e1, e2, e3, which is the identity matrix. 209 00:15:34,980 --> 00:15:39,700 And so what I end up having to do is solve three systems of equations. 210 00:15:39,700 --> 00:15:47,950 I have to solve Ax1 equals e1, Ax2 equals e2, and Ax3 equals e3. 211 00:15:50,420 --> 00:15:55,700 So, if I was actually computing the inverse, remember, before I said I could 212 00:15:55,700 --> 00:16:00,660 compute the solution to Ax equals B by pre-multiplying by A-inverse. 213 00:16:00,660 --> 00:16:03,720 So if I already knew what the inverse was, I could just 214 00:16:03,720 --> 00:16:07,999 multiply and find that, well, A-inverse B is going to be the solution. 215 00:16:10,970 --> 00:16:13,402 But if I just want to actually solve that 216 00:16:13,402 --> 00:16:16,640 system, I would just be solving one of these. 217 00:16:16,640 --> 00:16:19,650 If I want to find the A-inverse and use A-inverse to 218 00:16:19,650 --> 00:16:22,810 solve that system, I end up solving three of these. 219 00:16:22,810 --> 00:16:26,584 So it ends up being three times as much work to find a inverse to 220 00:16:26,584 --> 00:16:28,582 solve that problem as it would be 221 00:16:28,582 --> 00:16:32,164 just to directly solve that problem using elimination. 222 00:16:32,164 --> 00:16:36,194 So computing A-inverse three times 223 00:16:36,194 --> 00:16:40,180 as much work as solving Ax equals b. 224 00:16:40,180 --> 00:16:42,628 So the, this is going to be sort of one of the 225 00:16:42,628 --> 00:16:45,552 morals of the story here is quite often if you need to 226 00:16:45,552 --> 00:16:49,156 compute an inverse, if you see an inverse in a matrix formula, 227 00:16:49,156 --> 00:16:53,810 there's a better, faster way to solve the problem without using an inverse. 228 00:16:55,966 --> 00:16:59,303 so in this case, I, I was looking at a 3 x 3 case. 229 00:17:03,200 --> 00:17:06,332 And I said that you'd have to solve this three times. 230 00:17:06,332 --> 00:17:09,188 So it turns out that if you look at a 4 x 4 case or a 5 231 00:17:09,188 --> 00:17:13,472 x 5 case, the, the three times ends up being an upper bound on the amount of 232 00:17:13,472 --> 00:17:15,716 work you have to do because you've done 233 00:17:15,716 --> 00:17:18,844 enough elimination on A in solving these first three, 234 00:17:18,844 --> 00:17:21,564 that you can kind of recycle that you know, 235 00:17:21,564 --> 00:17:25,320 if you're are intelligent about the algorithm you use. 236 00:17:25,320 --> 00:17:28,395 And so the algorithm is called the Gauss-Jordan 237 00:17:28,395 --> 00:17:33,570 method, which I'm not going to go through this here but it's essentially just 238 00:17:33,570 --> 00:17:38,800 doing this elimination problem and you can do this in n cubed elimination steps. 239 00:17:40,610 --> 00:17:44,978 But if you compare that to solving Ax equals b, this requires n 240 00:17:44,978 --> 00:17:50,290 cubed divided by 3 or roughly n cubed divided by three elimination steps. 241 00:17:50,290 --> 00:17:51,660 So you're going to be doing less work. 242 00:17:53,522 --> 00:17:58,360 sorry, not n cubed. This is not quite right. 243 00:17:58,360 --> 00:18:02,188 Not elimination steps, but mathematical operations. 244 00:18:02,188 --> 00:18:05,408 but the point I want you take away from this 245 00:18:05,408 --> 00:18:08,536 is, it's 1 3rd as much work to solve Ax 246 00:18:08,536 --> 00:18:12,216 equals b using elimination as it would be to find 247 00:18:12,216 --> 00:18:16,130 an inverse and then use that to solve Ax equals b. 248 00:18:21,060 --> 00:18:23,104 And I remember I said if I, if I was 249 00:18:23,104 --> 00:18:26,097 doing elimination and I ran into a row that was all 250 00:18:26,097 --> 00:18:28,944 zeroes, I couldn't find a pivot in that row so 251 00:18:28,944 --> 00:18:32,530 that particular system does not have a full set of pivots. 252 00:18:34,700 --> 00:18:35,380 Oops. 253 00:18:35,380 --> 00:18:38,090 And now, I have this idea of invertibility. 254 00:18:38,090 --> 00:18:41,790 It says some matrices have inverses and some matrices don't. 255 00:18:41,790 --> 00:18:46,434 So if a matrix has an inverse that I can't find a non-zero vector 256 00:18:46,434 --> 00:18:51,290 that has the property that if I say a times that vector, I get zero. 257 00:18:53,140 --> 00:18:56,420 And so it turns out these two things are related, 258 00:18:56,420 --> 00:18:59,780 so let's supposed A is a square matrix and suppose 259 00:18:59,780 --> 00:19:01,220 there are n pivots. 260 00:19:02,260 --> 00:19:05,460 Then I could solve each one of these equations and that would 261 00:19:05,460 --> 00:19:08,580 give me these xis that would be the columns of my inverse. 262 00:19:08,580 --> 00:19:12,948 So I can put the, these column vectors xi together 263 00:19:12,948 --> 00:19:18,010 to make an matrix that's the inverse of my matrix A. 264 00:19:23,090 --> 00:19:28,130 And so elimination then gives a complete test for A-inverse to exist. 265 00:19:28,130 --> 00:19:30,270 So there has to be n pivot. 266 00:19:30,270 --> 00:19:33,970 So, this isn't a complete proof. This is just the result. 267 00:19:33,970 --> 00:19:37,150 So in a, a Linear Algebra textbook, they will tell you exactly how to, 268 00:19:37,150 --> 00:19:41,330 how to do this because it's sort of something you have to do both directions. 269 00:19:41,330 --> 00:19:44,294 But essentially if a matrix A is invertible, 270 00:19:44,294 --> 00:19:46,620 then you can find a set of n pivots. 271 00:19:46,620 --> 00:19:49,180 So you say, it has a full set of pivots. 272 00:19:49,180 --> 00:19:52,512 And if a matrix is non-singular, that means every row has 273 00:19:52,512 --> 00:19:55,790 a pivot then that matrix is also going to be invertible.