 Hi, I'm Zor. Welcome to Unisor Education. I would like to spend some time solving certain problems related to inverse matrices. And as usually in case of problems, I encourage you to go directly to Unisor.com website and notes for this lecture contains basically all these problems in a written form. And please try to solve them all just by yourselves. I think it's very important to spend some time at least to familiarize yourself with the problem even if you cannot really solve it. But these problems are really very very simple and I'll spend a very very short period of time to present them. Alright. Problem number one, actually most of them are like this, they are direct consequences from the definitions of inverse matrix and others. So problem number one, I have to prove this. That inverse matrix to the identity matrix is itself, this same identity matrix. Well, let me first just give a simple example. I mean obviously if you have let's say a two-dimensional identity matrix and multiply it by itself you will get, okay the first row first column of the result is product of first row vector times first column vector which is one times one plus zero times zero which is one. Now the element one two first row second column, okay so it's first row second column, one times zero plus zero times one which is zero. Now two one, second row first column, second row first column zero times one plus one times zero is zero. And element two two which is second row and second column zero times zero plus one one, that's one. So obviously we have the result of multiplication of matrix of this matrix by this is identity which means that this is inverse of this. Now this is identity and this is identity and this is identity so that actually the illustration of this particular rule that inverse to identity is itself. How can I prove it? Well to prove it I have to, well to prove that this is an inverse of this I have to prove this and I have to prove this right? So multiplication of my original matrix by something which I consider might be an inverse on the right or on the left should be equal to identity. Well if I claim that inverse to identity is itself I actually have to have this equality and this. Now if I think that these are equal then that should be equal and these are indeed true identities. Why? Because that's the definition of the identity matrix, right? So if I multiply this by this or this by this left or right I will get an identity which means one is inverse of another by definition of the inverse, right? So as I said it's a direct consequence of the definition. Okay next one what if I will inverse matrix twice? So consider matrix is invertible so from the matrix U I get this one which equals to identity matrix with all diagonal elements equal to one and I know this. Again this is a definition of inverse matrix and I assume that my matrix U is invertible which means inverse matrix does exist. Now what if I will try to inverse it again? Now if I want to inverse again now if I multiply it by something which I consider to be an inverse of this now what should I get? Well I should get identity, right? That's the properties, that's the definition, I don't need these parenthesis that's the definition of the inverse matrix but look at this, this is exactly what I'm looking for. If instead of A I will substitute U I will get this. I know that U and U to the minus one are that this is inverse of this which means these are true identities which are given to me. So that means that A and U are exactly the same if you compare these two and if you compare these two you will see that A and U are exactly the same again. So this inverse matrix to the inverse matrix is actually the original matrix itself and again it's a direct consequence from the definition. Well you see that these are really kind of trivial things but it's interesting to discuss these are true statements. Now the third one is not really a direct consequence of the definition and it's actually quite interesting. If you have a product of two matrices and you would like to inverse it and let's consider that these matrices are invertible by themselves each of them. Then it's equal to multiplication of their inverse but in inverse order. So to inverse the product you have to inverse each component but multiply it in reverse order. Again, how can I prove that this is x times y inverse? Well I have to multiply this by x times y and check if it's equal to identity matrix. So let's multiply xy by this one and let's see what it's equal to. If it's equal to one, I mean if it's equal to identity, sorry and multiply on another side on the left it's also equal to identity. Then this is the proof that this is inverse of this. So let's just check. Alright, so I multiply on the right. Now multiplication of the matrices as you know is associative. We already passed this in the lecture dedicated to how to multiply matrices. So the associativity is a true property of the matrix multiplication, not commutatively. So associative but not commutative. But we don't need commutativeness here, we need only associative law. By associative law I can just open the parenthesis and put parenthesis in any way how I want it. So I wanted to do it this way. Times y times y minus one times x minus one. That's where it is. Now by definition of the y to the minus one this is by definition an inverse matrix to y, which means their product is identity. Now identity being multiplied by x gives me x and x times x to the minus one, again by definition of the inverse matrix is equal to identity. Now if I will do in a reverse order y minus one x minus one x y is equal to again I will open the parenthesis and I will put it in this way. Now this is identity. Multiply by minus one gives me anything and then this is identity as well. So no matter how I multiply this on the left or on the right of the x times y I will give identity. Which means that this is x times y inverse. Right? So this is really a little bit less trivial and it looks interesting. Now what's also interesting is this. It can be very easily expanded by induction to a product of n matrices. Assuming again that each matrix has exactly the same dimension square matrix n times n and each matrix is invertible then this is a true statement. To invert the product you have to invert each component and multiply the results in the reverse order. Now this can be proven by induction it's absolutely trivial so I'm not going to stop on it. I do suggest by the way yourself to do this particular small exercises will take you another couple of minutes. Now certain matrices are difficult to invert. Well with dimensional matrix in the previous lecture I actually explicitly built the inverse matrix. Now in case of three dimensional and more dimensionalities it's a little bit more difficult. However there are certain matrices which are very easy to invert regardless of dimensionality and these are diagonal matrices. So diagonal matrices is the matrix with only diagonal elements not equal to zero. Everything else is equal to zero. So D1, D2 and D3 are not equal to zero. And everything else now this has coordinates 1 1 first row first column second row second column third row third column or 4 5 etc. To invert that matrix is very easy. That's what it is. So you invert each number and they are not equal to zero put it in exactly the same position and the result would be the inverted matrix. Well to prove it is very actually easy. If you want to know what is the result of the multiplication of this times this. Now let's think about the matrix which is the result. Well it's a 3 by 3 matrix obviously. Now the coefficient, the element of this matrix at row i column j would be the result of the vector row multiplied as a scalar multiplication, scalar product of the column number j. So row number i by column number j. Now only if i and j are the same like second row and second column only in this case we will have a bunch of zeros and one non-equal to zero element the element number equal to the row number. The second row so the second element will be multiplied by the corresponding column with exactly the same property. So only one element which is exactly on the place equal to the column number would be not equal to zero. All others will be equal to zero here and there. So only elements like second row and second column first row and first column. Only these elements will be not equal to zero and precisely equal to one because i multiply d2 by 1 over d2, right? Or g1 times 1 over d1. All other pairs of vectors, let's say third row and the first column, they have these non-zero elements in different places which means i multiply zero by something or something by zero. So the result is zero present in every component if i multiply as a scalar product this vector by this vector. So only second by second or first by first or third by third multiplication would give me the result equal to one and the rest would be zero and that's exactly what identity matrix actually is. So to invert the matrix which has only these elements along the main diagonal non-equal to zero, you just have to invert each element and place it in the same position. So these are diagonal matrices. That's the general term. Diagonal matrices are very easy to invert. You just invert each element by itself and put it in place. Now another very easy problem is this. If you want to invert the matrix multiplied by a scalar, by a constant now what is this? Well for obvious consideration is this. Now why? How can I prove it? Well if i multiply Ka times one over Ka minus one, now multiplication of matrices is not commutative but multiplication of matrix and constant actually is commutative. So first I will open all the parenthesis using the associative law. I put the parenthesis here then I invert them, then I put parenthesis again. This is one and this is identity matrix. So that's how you prove it. And the last problem which I wanted to present to you is this one. Operation of transposition when we are reflecting the matrix around the main diagonal. So main diagonal stays the same but all other elements are changing the places. So if I have a matrix A I transpose into matrix B, the rule is B i j equals to A j i. Row and columns are changing places, right? So we were talking about matrix transposition when we were talking about the basic operations with matrices. Now one of the properties which matrix transposition actually has is very much similar by the way to inversion. If you would like to transpose the product of two matrices then you transpose each one of them but multiply in an opposite order. It's very easy to prove. I would like actually you to spend some time to prove it. It's just another couple of minutes and more than that. And it's all based on the fact that the transpose matrix i j has the same element as original matrix j i. Now using this I can actually prove this very easily. Now how can I prove that this is inverse of ut transposed U? Well I have to multiply this times this like if it's equal to identity. Same thing on both sides. Well let's think about. Now U transposed times something else transposed according to this it's equal to original this original this and transposed. Now but this is identity because U minus one is an inverse of U. Now transposed transposition of identity matrix is the same identity matrix because it's symmetrical. You have all zeros above the main diagonal and all zeros below main diagonal and main diagonal stays in place. If it has only ones on the main diagonal and all zeros around it then transposition will give you exactly the same thing. Now if I do it on another side so I multiply this on the left of this. Same thing I will use exactly the same property. It's transposition of this times transposition of that. So I will use reverse order and then transpose. And again this is identity matrix and transposed will give you identity matrix. Well that's it. Well let me finish this particular lecture with again referring you to Unisor.com where I would suggest you to go through notes for this particular lecture again. All these problems are there and just try to do it yourself. I mean after you have listened to this lecture you should have absolutely no problems to do it yourself. And if you will be on that side registered then you can go through the whole educational process with your supervisor or parent. They can enroll you in certain topics. You can take exams. You can take as many times this exam as you want and basically present the results to your supervisor or parent. I do suggest you to get engaged in this educational process. It's very very interesting and important for development. Well that's it. Thank you very much and good luck.