Matrix Expansion Method
Matrix expansion is a mathematical technique for solving linear equations. It is based on the idea that a certain number of unknowns can be eliminated if coefficients of each variable are expanded into a matrix. This technique is useful for solving systems of linear equations with a large number of variables, as it can reduce the time and effort required to find a solution.
The matrix expansion method depends on the fact that an equation can be written as an augmented matrix. An augmented matrix is a rectangular array of numbers which consists of the coefficient matrix of the system of equations, along with the right-hand sides of the equations. By expanding the matrix into an augmented form, it is possible to quickly obtain solutions to the system of linear equations, as it eliminates the need to individually solve each equation.
To use the matrix expansion method, the coefficients of each variable must be written down in a matrix form. This means that, to solve a system of three linear equations, the coefficients of each of the three variables should be arranged in a 3x3 matrix. In addition, a column vector should also be created, which contains the right-hand-side values of each equation. The augmented matrix should then be expanded by multiplying each row by a different number.
Once the augmented matrix has been expanded, it is then possible to find a solution to the system of equations. To do this, the matrix is put into reduced row echelon form, which allows for easier manipulation. From the row-echelon form, the values of the variables can then be determined by using simple operations on the augmented matrix.
The matrix expansion method can also be used for solving systems of non-linear equations. In this case, the coefficients of each variable should be written in a matrix form and the augmenting column vector should contain the right-hand-side values of the equations. The matrix can then be expanded by multiplying each row by a different number and equations can be solved as usual.
In conclusion, matrix expansion is a useful mathematical technique for solving both linear and non-linear equations. It involves forming an augmented matrix, expanding the matrix by multiplying each row by a different number, and finally putting the augmented matrix into reduced row echelon form to obtain the solution. This method can be useful for solving systems of equations with a large number of variables, as it reduces the time and effort required to determine the solution.
