[BOŞ_SES] Hello. We have seen some guidelines to calculate determinants. Now some of the basic rules for using these determinants We will obtain the properties. We use as a display for it; the rows of a matrix One, two, three, Ajman, in the Let's call. there are n rows, each row in the example j'nc line if we get the first line j is determined for the indicator to a constant stay, j will be number two on the diagonal, we are showing as jnr. Each line is this happening n-dimensional vector. To give an example, let's take a double binary matrix, one, two, three, you get four. The rows of this matrix we define the rule here is as follows: A, vector forming the first line, one and two. A two vectors forming the second line. We find it easy determinants of minus six minus two is four. Now we have the rules of the determinants We have found as follows: If we intend to collect on the lines i We stabilizes, jiu changing second indicators, According to the i stands for the i-th row select line. This rule. Some properties now using this rule saptayacağız determinantların. First we call uniformity, we call any Homogeneity If we hit the line with a t t multiplied by the determinant. How do we do that? This law of our determinant. ij'yl we will hit a big ij, they are counted from the cofactor, and will collect on-line, i will collect the jar are holding steady. We are writing our j'nc by line after the same rule here, but we no longer t'yle çarpılmış a ij'ler, t'yle çarpılmış a ij'ler. To have been affected by this collection t, we get t outside. When we get out the back, t times as you can see this happening is the product of a ij. This gives the determinant t times the mean. So when he hit a line t'yl t'yl determinant itself is multiplied. For example, let's do it. The first line in this line I multiply this matrix with UC, then elements will be three and six, Calculate the determinant of its twelve minus eighteen; minus six is going on. We find that this is the determinant of minus two, indeed three times, How fast till we hit the old determinants unmultiplied determinants. According to the linearity of the collection line now we say; we another in a row on a line wherein The line we have demonstrated in C using eklesek determinants will prove right now, determinant, as we shall prove, CA has become the determinant addition, a de j'nc row that C has been put in place state determinant. Proof of this is easy, we apply the same rule again on our air, our ij elements, A small product of big ij ij here. We put this in a C ij Turn that we put him here, C because it is a vector C, C two, there are components in Cn. We stood with it after the coefficient. Now we see that here now, This collection can be divided into two parts; The initial first determinant, starting in the second row determinants j'nc Instead of adding this added C line. Therefore, we find that we have achieved this result. Example; a brew of matrix, one, two, three, consisting of four Let's add a vector, for example minus the first two rows in a matrix. If we add here a three minus one becomes two, the determinant When we calculate twelve minus three; We find nine. This proved to us that we feature, the fact that this theorem, Locate says the determinant, Insert it to the first row of the determinant C. Now we're looking at here is minus two primary determinants, the second determinant of eight, minus, but we will change it a sign, plus three plus three to eight; eleven. This time we gather here two really feature We are giving the checksum, we are giving an example. If a row of a matrix of a determinant is zero, say j'nc line is zero, this determinant is zero. Proof of this is easy; We are looking at a ij'y to, This consists of j'nc the item, so I fixed upon collection jar. All this was zero, since the first zero vector is zero, zero second, third, zero, zero, zero, zero mean. Reset all of these cofactors which means we stood, of course, these cofactors Each product we find the sum of zero for zero will give zero. Example, providing a numeric; it still our one, two, three, four matrix zero we put on the first line, We find it really zero minus zero determinant. Determinant of the matrix is equal to one unit, which is an important feature. What did he mean the unit matrix; one in each row have a rest zero. A number on the diagonal, so I j'nc the line if we take the unit matrix, where zero zero t that would be j'nc the item will have a rest zero, zero, zero will be. Now this time, we calculate the corresponding cofactors all on the line cofactors will be zero, but this is the first on a This j'nc the unit to line and column j'nc threw back the minus n minus bir'lik will remain a unit matrix. Here again we use here as he was in a a We meet with something, so that when we opened I ij is given here, everywhere zero, which is a single number, a cofactor that it also opposed. He is also a co-factor, means that there will be a once, the rest will be all zeros, we will find one. Example; Let's take two binary matrix unit, a once; A zero times zero; zero when the unit matrix We proved to be a determinant. Key features of these now we call determinants of them as propositions If we use the features we find we can get after that. Now we'll show you where the various properties for it, We'll admit that before we find the basis of four feature others will be able to prove it. For now, take a break, after eight feature We again proved but the differences between them, now formulated, this Laplace in the calculation chain, the algorithm We will prove them's all go on the basis of four key features we prove this. Here we give a break for it now,
