How can matrices be used to solve systems of equations

A matrix equation is an equation in which a variable stands for a matrix. You can solve the simpler matrix equations using matrix addition and scalar multiplication.

Matrix equations can be used to solve systems of linear equations by using the left and right sides of the equations. Since the matrix is essentially the coefficients and constants of a linear system, the three row operations preserve the matrix.

For example, swapping two rows simply means switching their position within the matrix. Also, when solving a system of linear equations by the elimination method, row multiplication would be the same as multiplying the whole equation by a number to obtain additive inverses so that a variable cancels.

Finally, row addition is also the same as the elimination method, when one chooses to add or subtract the like terms of the equations to obtain the variable. Therefore, row operations preserve the matrix and can be used as an alternative method to solve a system of equations. This matrix is then modified using elementary row operations until it reaches reduced row echelon form.

Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original. There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are Gaussian elimination and Gauss-Jordan elimination. By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix can be transformed to a row echelon form. This transformation is necessary for solving a system of linear equations.

A Matrix. Why does [x y z] go there? Because when we Multiply Matrices the left side becomes:. Then as shown on the Inverse of a Matrix page the solution is this:. It means that we can find the values of x, y and z the X matrix by multiplying the inverse of the A matrix by the B matrix. In this case, the solution either does not exist or there are infinitely many solutions to the system.

Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC. Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors.

Varsity Tutors connects learners with experts.