Inverting Matrices

Understanding Inverse Matrices

An inverse matrix is a concept exclusive to square matrices; it does not exist for matrices that are not square.
The inverse of a matrix A is represented as A^-1; when A is multiplied by A^-1 the result is the identity matrix (I).
Not all matrices have an inverse. A matrix without an inverse is called a singular or non-invertible matrix.
A matrix has an inverse if and only if its determinant (det(A)) is non-zero.

Calculating the Inverse Matrix

One way to calculate the inverse is by using the formula A^-1 = 1/det(A) adj(A), where adj(A) is the adjugate of A.
The adjugate of a matrix is the transpose of its cofactor matrix.
For a 2x2 matrix, [a, b; c, d], the inverse can be found as 1/(ad - bc)[d, -b; -c, a], where (ad - bc) is the determinant.
The process of finding the inverse for larger matrices can be complex and consists of finding the matrix of minors, then converting that into a cofactor matrix, and finally transposing it to get the adjugate.
Each element of the inverse matrix must then be divided by the determinant of the original matrix.

Inverse Matrices and Systems of Equations

Inverse matrices are used to solve systems of linear equations.
If A, X, and B are matrices, and A*X = B describes a system of linear equations, the solution can be found by X = A^-1 * B, using the inverse of matrix A.
However, if the determinant of matrix A equals zero, meaning A has no inverse, such methods cannot be applied. Other methods like Cramer’s rule or Gaussian elimination should be used instead.

Properties of Inverse Matrices

The inverse of a matrix is unique. Any given square matrix either has one unique inverse or none at all.
If a matrix A has an inverse, the inverse itself is also invertible, and (A^-1 )^-1 = A.
The inverse of the product of two invertible matrices equals the product of their inverses, in reverse order. If A and B are two invertible matrices, then (AB)^-1 = B^-1 * A^-1.
If a matrix is multiplied by a scalar, the inverse of the resulting matrix equals the inverse of the original matrix divided by the scalar. For a scalar k and an invertible matrix A, (kA)^-1 = 1/k * A^-1.