Finding the Inverse of a Matrix – Higher Advanced Mathematics SQA Revision

Introduction to Finding the Inverse of a Matrix

The inverse of a matrix, denoted as A⁻¹, is a unique matrix that, when multiplied with the original matrix A, results in the identity matrix.
The identity matrix, represented by the symbol I, is a special square matrix with ones on its main diagonal and zeros elsewhere.
An inverse matrix exists only for a non-singular matrix, a matrix whose determinant is not equal to zero.

It is only possible to find an inverse for a 2 x 2 or 3 x 3 matrix. Higher order matrices have much more complicated methods for finding inverses.
The inverse of a 2 x 2 matrix can be found using the formula: A = [[a,b],[c,d]] A⁻¹ = (1/det(A)) * [[d, -b],[-c, a]]
The determinant (det(A)) of a 2x2 matrix [[a,b],[c,d]] is given by ad-bc.
When finding the inverse of a 3 x 3 matrix, one needs to find the matrix of minors, then use this to create the matrix of cofactors, convert this to the adjoint matrix and then multiply by the reciprocal of the determinant.

Note that for two matrices A and B, the inverse of AB is the reverse product of the individual inverses, i.e., (AB)⁻¹ = B⁻¹ A⁻¹.
Also note that the inverse of the inverse of a matrix A gives back the original matrix, i.e., (A⁻¹)⁻¹ = A.
A matrix multiplied by its inverse always gives the identity matrix,i.e., AA⁻¹ = A⁻¹A = I.

The inverse of a matrix is frequently used in physics and engineering to describe systems of linear equations.
The inverse of a matrix can also be used to determine whether a matrix transformation is invertible, which is a key concept in the field of linear algebra.

It’s crucial to remember the formulas to calculate the inverse of a matrix as well as the properties and applications of the inverse.
Checking whether a matrix is singular or non-singular is the first step for finding the matrix’s inverse.