Solving systems of equations using matrices

A system of linear equations can be represented as matrices, which offers a structured and practical approach to finding the unknown variables.
When expressed as a matrix, the system of equations translates into the matrix equation Ax = b. Here, A is known as the coefficient matrix, x is the unknown variable matrix, and b is the constant matrix.
To solve this system, the inverse matrix of A, denoted as A^-1, is found. If the inverse exists, it can be used to solve for x in the equation by multiplying both sides of the equation by the inverse: x = A^-1b.
Finding the inverse involves several steps:
- First, create the augmented matrix [A I], where A is the original matrix and I is the identity matrix.
- Then, use row operations to transform the matrix into row-echelon form.
- Continue with the row operations until reaching the reduced row-echelon form [I A^-1].
- The matrix on the right-hand side of the vertical line is the inverse A^-1.
However, not all matrices have inverses. Matrices that don’t have inverses are known as singular. If you attempt to find the inverse and instead find a row of zeros, the matrix doesn’t have an inverse.
If a matrix doesn’t have an inverse, it implies one of the following: no solution to the system of equations exists, or that solutions are not unique, i.e., there are infinite solutions.
In such a case, row operations can still be used to solve the system of equations. We end up with a matrix in row-echelon or reduced row-echelon form, and this can be used to find the solution.

The determinant of a matrix is another useful concept. The determinant of a matrix A, denoted as det(A) or

, is zero if A is singular and non-zero if A has an inverse.

The determinant also provides insight into the system’s behaviour. If

= 0, the system has no solution or infinite solutions. If

≠ 0, the system has a unique solution.

The use of matrices simplifies the process of solving systems of equations, removing the need for substitution or elimination methods, especially when dealing with three or more unknowns.
However, due diligence and precision are crucial during matrix operations since one small error can lead to an incorrect solution.
Be aware of the limitations of the matrix method. It can only be used to solve linear equations and cannot handle nonlinear systems. The equations must also all be in standard form ax + by + cz = d.
Although matrices can be called an advanced topic in Core Pure Mathematics 1, understanding their use to solve systems of equations provides a strong foundation for more complex mathematical manipulations down the road.