Finding the determinant of a matrices

May 21, 2020 · Abstract. And the motivation is, because when you take the determinant of a 3 by 3 it turns out-- I haven't shown it to you yet-- that the property is the same. Determine the co-factors of each of the row/column items that we picked in Step 1. EVALUATING A 2 X 2 DETERMINANT If. The first way was to augment the ???n\times n??? matrix with the ???I_n??? identity matrix, and then put the matrix into reduced row-echelon form. Using the determinant to find the Area of a Triangle in the coordinate plane: We can find the area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) using the formula below. The determinant is |A| = a. Determine the sign of your answer. 4: Applications of the Determinant The determinant of a matrix also provides a way to find the inverse of a matrix. If anybody could explain the mechanics behind this first part So for an n × m n × m matrix, let k = min(n, m) k = min ( n, m) then compute all determinants of k × k k × k submatrices, perhaps with alternating sign. Then we completely row reduce, the resulting matrix on the right will be the inverse matrix. A 3x3 matrix has three columns and three rows, and therefore, 9 entries. haccks. Sep 17, 2022 · In order to find the determinant of a product of matrices, we can simply take the product of the determinants. 3 Find the determinants of the The determinant is the sum of product terms made up of elements from the matrix. patreon. answered Mar 13, 2014 at 16:35. For a 4x4 matrix, you expand across the first column by co-factors, then take the determinant of the resulting 3x3 matrices as above. To understand determinant calculation better input Determinant of a 3x3 matrix. Example, if A is 3x3, and Det (A) = 5, B=2A, then Det (B) = 2^3*5=40. It is calculated from the elements of a matrix using a special formula. However, for those brave souls that came here to learn to do the real work of calculating a determinant for ANY size matrix, the rule of Sarrus is only a stepping stone to one location – the determinants for 3×3 matrices. May 7, 2017 · Unfortunately this is a mathematical coincidence. In this section, we will learn the two different methods in finding the determinant of a 3 x 3 matrix. However, multiplying a row by some factor will lead to the determinant being multiplied by the same factor. This isn't too hard, because we already calculated the determinants of the smaller parts when we did "Matrix of Minors". DETERMINANT OF A 3 X 3 MATRIX . We want to find that determinant. Disaster for invertibility. Mar 12, 2009 · Thanks to all of you who support me on Patreon. Inverse of a Matrix using Minors, Cofactors and Adjugate; Use a computer (such as the Matrix Calculator) Conclusion. 105k 26 179 269. Here, we’ll focus on LU decomposition. This minor is given by. As long as long as you are looking at a matrix in the form of x by x, where both values of x are equivalent, it is possible to find the determinant. then. Now it's clear that the first two columns are the same, and that means that the determinant must be 0 0. minor(A)12 = det[4 3 2 1] = −2. println("v:\n" + v); I'm not really sure if this is the right way to approach this problem or if I just have some mistakes within my code. Recall that when working with large matrices, Laplace Expansion is effective but timely, as there are many steps involved. Using this fact, we want to create a triangular matrix out of your matrix \begin{bmatrix} 2 & 3 & 10 \\ 1 & 2 & -2 \\ 1 & 1 & -3 \end{bmatrix} So, I will start with the last row and subtract it from the second row to get \begin{bmatrix} 2 & 3 & 10 \\ 0 & 1 & 1 \\ 1 & 1 & -3 \end{bmatrix All rotation matrices have unit determinant; since , it cannot be a rotation matrix: Show that the matrix is orthogonal and determine if it is a rotation matrix or includes a reflection: Up to the input precision, , which shows that is orthogonal: A block matrix (also called partitioned matrix) is a matrix of the kind where , , and are matrices, called blocks, such that: and have the same number of rows; and have the same number of rows; and have the same number of columns; and have the same number of columns. We show how the determinant of a matrix can be calculated using complex analysis of one variable. Suppose we are given a square matrix [latex]A[/latex] where, Nov 21, 2023 · A matrix is a two-dimensional array of numbers (which can be real or complex), arranged in columns and rows. The identity matrix doesn't change the plane at all, so it stands to reason that it has a determinant of 1. It maps a matrix of numbers to a number in such a way that for two matrices #A,B# , #det(AB)=det(A)det(B)# . $\endgroup$ – The determinant of A, denoted as |A| ∣A∣ or \det (A) det(A), is calculated as follows: Step 1: Multiply the elements of the main diagonal (from top left to bottom right): a \cdot d a ⋅ d. Multiply the row/column items from Step 1 by the appropriate co-factors from Step 2. 1, 3. Step 2: Multiply the elements of the second diagonal (from top to bottom): b \cdot c b ⋅ c. Aug 8, 2022 · Multiply this by -34 (the determinant of the 2x2) to get 1*-34 = -34. To find the rank of a matrix, we can use one of the following methods: Find the highest ordered non-zero minor and its order would give the rank. Hence, det B = − det A ⇒ 16 = − ( − 16) Alright, so instead of switching rows, let’s scale a row in matrix A and see what happens. So let's say we have the matrix, we want the determinant of the matrix, 1, 2, 4, 2, minus 1, 3, and then we have 4, 0, minus 1. The determinant of a 3 x 3 Other Math questions and answers. either. Using: Elements of top row: 3, 0, 2 Minors for top row: 2, 2, 2. You might need: Calculator. The determinant can be a negative number. They help to find the adjoint, inverse of a matrix. It can be defined only for the matrices that have the same number of rows and columns. (a) [1032] (b) [0032] (c) [4632] Exercise3. Sep 17, 2022 · Put these two ideas together: given any square matrix, we can use elementary row operations to put the matrix in triangular form,\(^{3}\) find the determinant of the new matrix (which is easy), and then adjust that number by recalling what elementary operations we performed. float determinant(int m, float b[][m]); and the call should be like. That if the determinant of this is 0, you will not be able to find an inverse. We also describe the connection between unitary matrices and the problem of holomorphic isometries in complex geometry. out. In this section, we look at two examples where row operations are used to find the determinant of a large matrix. Each product term consists of n elements from the matrix. Google Classroom. It should be noted that there are other techniques used for remembering how to calculate the determinant of a \(3\times 3\) matrix. Although any choice of row or column will give us the same value for the determinant, it is always easier to When you go to find the determinant, remember that there were elements from the original 4×4 matrix that were times each of those 3×3 determinants. When it comes to matrices, beyond addition, subtraction, and multiplication, we have to learn how to evaluate something called a determinant. 3: Finding Determinants using Row Operations In this section, we look at two examples where row operations are used to find the determinant of a large matrix. To find the determinant of matrices, the matrix should be a square matrix, such as a determinant of 2×2 matrix, determinant of 3×3 matrix, or n x n matrix. float det = determinant(m, arr); edited Mar 13, 2014 at 16:48. Let’s take a look at the definition of the determinant: The determinant of a matrix is a scalar value that results from certain operations with the elements of the matrix. 1. (You can/should stop at 3 × 3 3 × 3, at which point it's easy enough to compute the final result manually. Introduction to Matrices: htt Mar 8, 2024 · The determinant is a term that is used to represent the matrices in terms of a real number. The determinant of an n x n square matrix A, denoted |A| or det (A) is a value that can be calculated from a square matrix. Each product term includes one element from each row and one element from each column. Matrices with a determinant of 0 will squash the plane down into a line or a point, thereby multiplying all areas by 0. Further to solve the linear equations through the matrix inversion method we need to apply this concept. N m 2M AaHdreM Bw2iJt1hb LIon afPi Onoi et QeK GAjl8gIe jb Hrfa Q t2 6. Det (kA)=k^n*Det (A). We find the determinant by computing the cofactor expansion along the first row. 1 Determinant of cartan matrix by elementary row /column operations Jul 1, 2024 · Determinant Expansion by Minors. −2 + 1 = −1 − 2 + 1 = − 1. Determinant of any square matrix of order 2×2, 3×3, 4×4, or n × n, where n is the number of rows or the number of columns. A system of linear equations can be solved by creating a matrix out of the coefficients and taking the determinant; this method is called Cramer's Sep 17, 2022 · The determinant of a square matrix is a number that is determined by the matrix. Multiply the main diagonal elements of the matrix - determinant is calculated. Notice that by switching two rows means we multiply the determinant of A by -1. Finding the Determinant of a Matrix via Complex Analysis Shan Tai Chan and Yuan Yuan Abstract. The determinant of a 3 x 3 matrix is a scalar value that we get from breaking apart the matrix into smaller 2 x 2 matrices and doing certain operations with the elements of the original matrix. Khan Academy is a nonprofit with the mission of providing a free Jun 3, 2023 · The determinant of the “new matrix,” which we call matrix B, is, det B = ( 3) ( 6) − ( 2) ( 1) = 16. 2, and 3. Convert the matrix into echelon form using the row/column operations. 3. This will shed light on the reason behind three of the four defining properties of the determinant, Definition 4. Long story short, multiplying by a scalar on an entire matrix, multiplies each row by that scalar, so the more rows it has (or the bigger the size of the square matrix), the more times you are multiplying by that scalar. Khan We have two options for finding the area of a triangle by using determinants: We could treat the triangles as half a parallelogram and use the determinant of a 2 × 2 matrix to find the area of this parallelogram, or we could use our formula for the area of a triangle by using the determinant of a 3 × 3 matrix. 4 2. 4 illustrate how row operations affect the determinant of a matrix. The first method is the general method. Add all of the products from Step 3 to get the matrix’s determinant. Jan 2, 2024 · To find the determinant, we normally start with the first row. Determinants. Answer. 1 Find the determinants of the following matrices. INTRODUCTION. Take the transpose of the cofactor matrix to get the adjugate matrix. It is NOT the case that the determinant of a square matrix is just a sum and difference of all the products of the diagonals. We rewrote those first two columns. The inverse of A is A-1 only when AA-1 = A-1 A = I; To find the inverse of a 2x2 matrix: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc). 2 and 3. Applying property 3 of Theorem 3. Then the number of non-zero rows in it would give the rank of the matrix. org/math/precalculus/precalc-matrices/inverting_matrices/e/matrix_determi This is our definition of the determinant of a 3 by 3 matrix. Which you use depends on where the element was placed in the 3x3 matrix. Calculate the determinant of 2×2 minor matrices. To calculate the determinant of a 3 × 3 matrix, recall that we can use the cofactor expansion along any row using the formula d e t ( 𝐴) = 𝑎 𝐶 + 𝑎 𝐶 + 𝑎 𝐶, where 𝑖 = 1, 2, or 3, and along any column. This is a new c However, if you were to find a matrix in the form of 2 by 3, 3 by 5, etc, it would be impossible to find the determinant. It is also a crucial ingredient in the change-of-variables formula in multivariable calculus. The first one was -2 and the second one was +2. In this article, you will learn about the adjoint of a matrix, finding the adjoint of different matrices, and formulas and examples. Jul 13, 2015 · Practice this lesson yourself on KhanAcademy. For a 1 x 1 Matrix. (2 1 −1 −1) ( 2 − 1 1 − 1) First note that the determinant of this matrix is. This is a determinant of a matrix of matrices, and they treat it like it is a 2x2 matrix determinant (and keep the det () operation after, which is even more confusing). ) Apr 25, 2017 · This precalculus video tutorial explains how to find the determinant of 3x3 matrices and 2x2 matrices. The main diagonal is the set of entries that run from the upper left-hand corner of the matrix down to the lower right-hand corner of the matrix. 8 is called a Vandermonde matrix, and the formula for its determinant can be generalized to the case. Enter the matrix. May 6, 2023 · As we saw, calculating the determinant of a 2×2 matrix is quite simple. For an n × n matrix M, the determinant of M (sometimes written | M |) is given by: det M = ∑ σ sgn(σ)m1 σ ( 1) m2 σ ( 2) ⋯mn σ ( n) The sum is over all permutations of n. the determinant of A (or "det A "): In other words, to take the determinant of a 2×2 matrix, you follow these steps: Multiply the values along the top-left to bottom-right diagonal. Multiply the values along the bottom-left to top-right diagonal. Also known as "Laplacian" determinant expansion by minors, expansion by minors is a technique for computing the determinant of a given square matrix . You are encouraged to research this rich topic. Determinant is defined only for square matrices. So 1, 2, 2, minus 1, 4, 0. 126 Determinants (a) minor (A)11 (b) minor (A)21 (c) minor (A)32 (d) cof (A)11 (e) cof (A)21 (f) cof (A)32Exere se 3. So by the Rule of Sarrus, we can rewrite these first two columns. Things to keep in mind: Determinant only exists for a square matrix. For a 1 x 1 matrix ( 1 row and 1 column )=>. A = [ 0 3 5 5 5 2 3 4 3] What is the determinant of A ? Show Calculator. Exercise 3. The determinant of a matrix has various applications in the field of mathematics including use with systems of linear equations, finding the inverse of a matrix, and calculus. Example 2. T his calculation, however, becomes more complex for 3×3 matrixes or higher. To calculate a determinant you need to do the following steps. The matrix in Example 3. When you get an equation like this for a determinant, set it equal to zero and see what happens! Those are by definition a description of all your singular matrices. This method requires you to look at the first three entries of the matrix. For each element of the first row or first column get the cofactor of those elements and then multiply the element with the determinant of the corresponding cofactor, and finally add them with alternate signs. Worst case scenario. 2. Formulate the matrix of cofactors. Jun 12, 2024 · Determinant of a Matrix is defined as the sum of products of the elements of any row or column along with their corresponding co-factors. determinant is 13. Feb 8, 2021 · Upper triangular matrices are matrices in which all entries below the main diagonal are 0. Lower triangular matrices are matrices in which all entries abo. 1. If is an matrix, forming means multiplying row of by . In previous lessons, we learned two ways to find the inverse matrix. If A and B are n × n matrices, then det (AB) = det (A) det (B). The number of product terms is equal to n! (where n! refers to n factorial ). If the determinant is not a whole number, you Step 4: Multiply by 1/Determinant. Next, you'll multiply your answer either by 1 or by -1 to get the cofactor of your chosen element. Determinants are the scalar quantities obtained by the sum of products of the elements of a square matrix and their cofactors according to a prescribed rule. Find the following. The cofactor is calculated by multiplying the element by the determinant of the submatrix formed by excluding the row and column of that element, and then alternating the sign. Oct 5, 2020 · In this video I demonstrate how to find the determinant of a 5 x 5 matrix by using the co-factor expansion then for the remaining 3 x 3 matrix I demonstrate The determinant of a 3x3 matrix shortcut method is a clever trick which facilitates the computation of a determinant of a large matrix by directly multiplyin Jul 20, 2015 · A very important property of the determinant of a matrix, is that it is a so called multiplicative function. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Mar 19, 2023 · By Definition 11. You da real mvps! $1 per month helps!! :) https://www. This procedure is illustrated in the third screen. Ideally, a block matrix is obtained by cutting a matrix vertically and A determinant is a property of a square matrix. Example \(\PageIndex{5}\): The Determinant of a Product Jul 28, 2023 · Definition. The determinant of a 2 x 2 matrix A, is defined as NOTE Notice that matrices are enclosed with square brackets, while determinants are denoted with vertical bars. com/patrickjmt !! Finding the Determinant of To find the inverse of a matrix, we write a new extended matrix with the identity on the right. For each element in the chosen row or column, multiply it by its corresponding cofactor. Always. Sep 17, 2022 · First we will find the determinant of this matrix. . To find a 3x3 determinant with no zeros, you have to find three 2x2 Sep 17, 2022 · The determinant of an upper-triangular or lower-triangular matrix is the product of the diagonal entries. If you subtract the third column from the first one, which is a valid transformation with respect to the determinant (it will leave it unchanged), you will get: ⎡⎣⎢1 0 4 1 0 4 3 −2 1 ⎤⎦⎥. Nov 24, 2017 · You want to look at gaussian elimination as this is commonly used to find the determinant of a squared matrix in computing. 2. LU Decomposition to Calculate the Determinant of a Matrix Nov 25, 2020 · Matrix z = new Matrix(new double[][]{{1, 2, 3},{0, 4, 5},{1, 0, 6}}); Matrix v = z. The focus of this article is the computation The determinant of a ends up becoming a, 1, 1 times a, 2, 2, all the way to a, n, n, or the product of all of the entries of the main diagonal. ) Extra points if you can figure out why. The determinant is a numerical value (positive or negative) calculated from the elements in a matrix and is used to find the inverse of a matrix; You can only find the determinant of a square matrix; The method for finding the determinant of a matrix is given by: Note that the determinant of a lower (or upper) triangular matrix is the product of its diagonal elements. This tool calculates determinants for matrices of arbitrarily large size. The value of the determinant has many implications for the matrix. Now find the determinant of the original matrix. determinant(); System. Also, the matrix is an array of numbers, but its determinant is a single number. A determinant of a transformation matrix is essentially a scaling factor for area as you map from one region to another region, or as we go from one region to the image of that region under the transformation. 2) For square matrix A, det(A) ≠ 0 iff the system has a unique solution. (Actually, the absolute value of the determinate is equal to the area. Set the matrix (must be square). Since we have a diagram with the Mar 26, 2016 · To evaluate the determinant of a matrix, follow these steps: If necessary, press [2nd] [MODE] to access the Home screen. (hint: to rotate a vector (a,b) by 90 A determinant is a property of a square matrix. Jul 13, 2015 · Using matrices to figure out if some combination of 2 vectors can create a 3rd vectorPractice this lesson yourself on KhanAcademy. The steps required to find the inverse of a 3×3 matrix are: Compute the determinant of the given matrix and check whether the matrix invertible. With the help of the determinant of matrices, we can find useful information of linear systems, solve linear systems, find the inverse of a matrix, and use it in calculus. It is however vector-valued, not real-valued, except for the square case. Points to note from Equations 2: Jun 4, 2024 · Given the following definition of a square matrix A, find the determinant for all sizes of the matrix. org right now: https://www. First, add \(-3\) times the first row to the second row. f 3 fA 2l2lF CreiEgHhQtRsJ 2r oe rs re Gr Fv je hdg. Each summand is a product of a single entry from each row, but with the column numbers shuffled by the permutation σ. Determinants originate as applications of vector geometry: the determinate of a 2x2 matrix is the area of a parallelogram with line one and line two being the vectors of its lower left hand sides. There are several ways to find the determinant of the matrix in JavaScript which are as follows: Different row-operations affect the determinant of the matrix differently. det(λI −Acl) = det(λ2I + (λ + 1)kLe)) = 0 d e t ( λ I − A c l) = d e t ( λ 2 I + ( λ + 1) k L e)) = 0. However, from matrix to matrix in that form, finding the determinant varies. Determinants can be used to find out the inverse of a matrix. Let’s practice this. The determinant is also multiplicative. Choose any row or column of the matrix. Sep 17, 2022 · Theorems 3. And to figure out this determinant we take this guy. 4. As a base case, the value of the determinant A matrix having m rows and n columns is called a matrix of order m × n or m × n matrix. 4. 2, we can take the common factor out of each row and so obtain the following useful result. ©l R2w0i1 T2q yK lu RtBaJ wSGo if st 9wia 6rBe J mLJL lC B. In addition, many modern calculators and computer algebra systems can find the determinant of matrices. Its is compatible to your function parameter if you will declare it as. If you need a refresher, check out my other lesson on how to find the determinant of a 2×2. 1, this is the determinant of the 2 × 2 matrix which results when you delete the first row and the second column. A square matrix is invertible if and only if det (A) ≠ 0; in this case, det (A − 1) = 1 det (A). To compute the determinant of an \ (n\times n\) matrix, we need to compute \ (n\) determinants of \ ( (n-1)\times (n-1)\) matrices. Sep 20, 2016 · 1) A system of linear equations has either no solution, a unique solution, or infinitely many solutions. Apr 20, 2015 · The determinant can be written as the sum of the product of the elements in the top row with their associated minors: so this determinant would be: a1,1 ∗⎡⎣⎢⎢⎢⎢⎢⎢a2,2 b2,3 ⋮ b2,n 0 a3,3 ⋱ ⋯ ⋯ ⋱ ⋱ bn−1,n 0 ⋮ 0 an,n⎤⎦⎥⎥⎥⎥⎥⎥ a 1, 1 ∗ [ a 2, 2 0 ⋯ 0 b 2, 3 a 3, 3 ⋱ ⋮ ⋮ ⋱ ⋱ 0 b 2, n ⋯ Mar 15, 2023 · The value of the determinant of a matrix can be calculated by the following procedure –. In this lesson, we will look at the formula for a $ 3 \times 3 $ matrix and how to find the determinant of a $ 3 \times 3 $ matrix. Sometimes there is no Sep 17, 2022 · In this section we give a geometric interpretation of determinants, in terms of volumes. Using Theorems 3. Feb 16, 2024 · All about the determinant of a matrix. khanacademy. The determinant for that kind of a matrix must always be zero. However, matrices can be classified based on the number of rows and columns in which elements are arranged. As another hint, I will take the same matrix, matrix A and take its determinant again but I will do it using a different technique, either technique is valid so here we saying what is the determinant of the 3X3 Matrix A and we can is we can rewrite first two column so first column right over here we could rewrite it as 4 4 -2 and then the second column right over here we could rewrite it -1 5 2. The determinant of a square matrix A is denoted by det A or | A |. Which is a super important take away, because it really simplifies finding the determinants of what would otherwise be really hard matrices to find the determinants of. The result generalizes both the determinant and the cross product. A determinant of 0 implies that the matrix is singular, and thus not invertible. A system of linear equations can be solved by creating a matrix out of the coefficients and taking the determinant; this method is called Cramer's Jul 27, 2015 · You have two main ways to find the determinant of a matrix 4 × 4: 1] Using the method of Laplace or of the Cofactors. Step 3: Subtract the product of step 2 (second diagonal Dec 4, 2016 · Hence its type will become float (*)[m]. Finding determinants of a matrix is helpful in solving the inverse of a matrix, a system of linear equations, and so on. Defining the determinant for 2x2 and 3x3 matrices. [ 1 1 3 0 0 − 2 4 4 1]. Similarly, minor(A)23 is the determinant of the 2 × 2 matrix which results when you delete the second row and the third column. 3) If you have a statement like x + y = 2 left after row-echelon form, this means that your system has infinitely many solutions. We can find the matrix determinant through different methods and algorithms in these cases. Press [ENTER] to evaluate the determinant. This method is good and easy to apply but very cumbersome! You must evaluate a lot of smaller determinants and it is possible, during these steps, to make mistakes (it also quite boring!!!): For example: 2] The second way is a This tool calculates the determinant of a matrix. Look at what always happens when c=a. For a 2 x 2 Matrix. 1 in Section 4. Although efficient for small matrices, techniques such as Gaussian elimination are much more efficient when the matrix size becomes large. The standard formula to find the determinant of a 3×3matrix is a break down of smaller 2×2determinant problems which are very easy to handle. It means the matrix should have an equal number of rows and columns. 2 Let A=⎣⎡10−2215431⎦⎤. E: Exercises Jul 26, 2019 · In this video I will teach you a shortcut method for finding the determinant of a 5x5 matrix using row operations, similar matrices and the properties of tri Jan 10, 2024 · The result is the determinant of the 3 × 3 matrix. Instead of memorizing the formula directly, we can use these two methods to compute the determinant. Adding a multiple of one row to another will not change the determinant. We end up with this calculation: Determinant = 3×2 − 0×2 + 2×2 = 10 Apr 15, 2020 · $\begingroup$ so $$\det(\alpha-a'A^{-1}b)$$ is the determinant of a constant, is the determinant of a constant just equal to a constant, couldn't find that in the properties of determinants in my textbook. Simply iterate until your determinant gets to reasonable size. We can solve linear systems with three variables using 3. Mar 4, 2017 · This video shows how to find the determinant of a matrix using microsoft excel. Press [ALPHA] [ZOOM] to create a matrix from scratch, or press [2nd] [ x–1] to access a stored matrix. 4, we can first simplify the matrix through row operations. Consider the following example. Subtract the second product from the first. The determinant is a single value, which is one of many numerical characteristics of a square matrix. 6. For any square matrix A, we have det (AT) = det (A). Oct 13, 2017 · Testing for a zero determinant. A = ± 1 2|x1 y1 1 x2 y2 1 x3 y3 1|, where the ± accounts for the possibility that the determinant could be Jul 21, 2013 · PC reduces an n × n n × n determinant to an (n − 1) × (n − 1) ( n − 1) × ( n − 1) determinant whose entries happen to be 2 × 2 2 × 2 determinants. It also doesn't satisfy 3. Reduce this matrix to row echelon form using elementary row operations so that all the elements below diagonal are zero. Determinant = -2 ( -16 ) + 2 ( -4 ) = 32 - 8 = 24. Here is another link which might be a good read. tx sx ye et jg ua xu lj yz nn