 In this video, I explain the solution of simultaneous linear algebraic equation by Gauss-Zodan method. Learning outcome. At the end of this session, the student will be able to solve the simultaneous algebraic equation by Gauss-Zodan method. Gauss-Zodan method, this is also a direct method and it is also called the modification of Gauss elimination method. I explain this method by considering three equations in the three unknowns that is a 1 x plus b 1 y plus c 1 z is equal to d 1, a 2 x plus b 2 y plus c 2 z is equal to d 2, a 3 x plus b 3 y plus c 3 z is equal to d 3. Now, quality is equation 1. The given system of equation can be put in the matrix from as a x is equal to the quality is 2, where a is the coefficient matrix, x is the column unknown matrix, b is the constant matrix here, a is equal to a 1, b 1, c 1 that is the first row, a 2, b 2, c 2, second row, a 3, b 3, c 3, third row, x is the column unknown matrix x, y, z, b is equal to d 1, d 2, d 3. The argumentary matrix a b, argumentary matrix it means that inserting or adding a column to a coefficient matrix which consists of the element of b, such matrices are called the argumentary matrix a b, that is equal to a 1, b 1, c 1, d 1, a 2, b 2, c 2, d 2, a 3, b 3, c 3, d 3. This method aims in reducing the coefficient matrix a to an diagonal matrix here. Now, step 1 use the element a 1 which is not equal to 0 to make the element a 2 and a 3 0 by elementary row transformation, that is argumentary matrix a b equal to a 1, b 1, c 1, d 1, 0, b 2 dash, c 2 dash, d 2 dash, b 3 dash, c 3 dash and d 3 dash here. Now, use the element of b 2 dash in the step 2 and make the element of b 3 dash and b 1 or becomes the 0 by using the elementary transformation, that is equivalent to a 1, 0, c 1 dash, d 1 dash, 0, b 2 dash, c 2 dash, d 2 dash, 0, 0, c 3 double dash, d 3 double dash here. Now, use the element c 3 double dash which is not equal to 0 to make the element c 1 dash and c 2 dash 0 by elementary row transformation that is equivalent to a 1, 0, 0, d 1 double dash, 0, b 2 dash, 0, d 2 double dash, 0, 0, c 3 double dash, d 3 double dash, call it is equation number 4. Now, step 4, the equivalent system of equations are the a 1 x is equal to d 1 double dash, from this x is equal to d 1 double dash upon a 1 dash, b 2 dash y is equal to d 2 double dash upon b 2 dash, c 3 double dash z is equal to d 3 double dash, this implies z is equal to d 3 double dash upon c 3 double dash, hence we get the required solution of the given equation 1. Now, first the video, reduce the following matrix into diagonal matrix, I hope all of you will get the result. Now, diagonal matrix means to reduce this matrix c to the diagonal matrix, these are the diagonal elements and above the diagonal are below the diagonal are make it to 0 by using the row elementary transformation. Now, by using the transformation r 2 equal to r 2 minus r 1 r 3 equal to r 3 minus r 1, that is 1 1 1 0 1 2 0 4, this is the first step. Now, to use the by the second step, the 1 which is not equal to 0, use this element, make the second element of the first row and the second element of the third row or make it to 0 by using the elementary transformation, that is r 1 equal to r 1 minus r 2 r 3 equal to r 3 minus r 2, that becomes 1 0 minus 1 0 1 2 0 0 2. Now, using the third element of the third rows and make the element of the third element of the first row and third element of the second row or becomes the 0, that is 1 0 0 0 1 0 0 0 2. Now, come to an example solve the system of equation by Gauss-Joran method, that is x plus 4 y plus 9 z is equal to 16, 3 x plus 2 y plus 3 z is equal to 18, 2 x plus y plus z is equal to 10 solution. The given system of equation can be written in the matrix form is the x is equal to b, where a is the coefficient matrix, that is a equal to 1 4 9 3 2 3 2 1 1 and x is the column unknown matrix x, y, z and b is equal to the column constant matrix, which is the retention element of the given system of equation, that is 16, 18 and 10 year. Augumentary matrix, that is a b is equal to 1 4 9 16 3 2 3 18 2 1 1 2 1 2 1 1 2 1 10. Now, reduce the coefficient matrix to a diagonal matrix, for that here first step is use the first element of the second row and the third row or to becomes 0 by using the first element of the first row, by applying the only row elementary transformation, that is R 2 is equal to R 2 minus 3 times of R 1, R 3 equal to R 3 minus 2 times of R 1, that is augmented matrix a b equal to 1 4 9 16 0 minus 10 minus 24 minus 30 0 minus 7 minus 17, for the our convenience, the we have to divide the R 2 by 2, because to get the smallest integer here, that for our convenience it is not a mandatory, that is R 2 is equal to minus R 2 divided by 2, that is a b equal into 1 4 9 16 0 5 12 15 0 minus 7 minus 17 minus 22. Now, then use the second element of the second row to make the second element of the first row and the second element of the third row or becomes the 0, the by using the row transformation, that is R 1 equal to the 5 times of R 1 minus 4 times of R 2, R 3 equal to 5 times of R 3 plus 7 times of R 2, that is augmented matrix a b equal to matrix 5 0 minus 3 20 0 5 12 15 0 0 minus 1 minus 5. Now, use the third element of the third element that is minus 1 to make the third element of the first row and second row will become 0 that is R 1 equal into R 1 minus 3 times of 2 equal into R 2 plus 12 times of R 3, that becomes the a b equal into 5 0 0 35 0 5 0 minus 45 minus 45 minus 45 minus 45 minus 45 0 0 minus 1 minus 5. Now, the equivalent system of equations are 5 x is equal to 35 that implies x is equal to 7, 5 y is equal to minus 45 that implies the y is equal to minus 9 minus z is equal to minus 5 that implies that z is equal to 5. Thus x is equal to 7 y is equal to minus 9 z is equal to 5 is the required solution of the given system of equations. References, numerical methods by B. S. Grevel