Inverse of a 2x2 Matrix

The Inverse of a 2x2 Matrix calculator computes the inverse of a 2x2 matrix.

INSTRUCTIONS: Enter the following:

• (A)  This is the 2x2 matrix

Inverse: The calculator returns the inverse of the 2x2 matrix.

The Math

An invertible matrix is a matrix that is square and nonsingular. In Linear Algebra, the inverse of a given matrix relates well to Gaussian elimination; you may wish to visit what it means to perform elementary row operations by going to our page on the Row Echelon Form of a 3x3 matrix. Although the Row Echelon Form of a `3times3` Matrix page does not directly relate to `2times2` matrices, performing elementary row operations is the same for all  `m times n` matrices. 

A square matrix is a matrix that has the same number of rows and columns, often times referred to as a `n times n` matrix. In the case above, we are taking the inverse of a `2times2` matrix, where there are two rows and two columns. A nonsingular matrix is a matrix whose determinant is not equal to zero; a singular matrix is not invertible because it will not reduce to the identity matrix. 

When it comes to the basic idea of an inverse, it is explained by Williams in the following manner (69):

Suppose you have two numbers such that
     `a*b=1` and `b*a=1`
this means that `a` and `b` are multiplicative inverses of each other. 

For example, take `a=frac(1)(5)` and `b=5.` It is clear that when you multiply `frac(1)(5) * 5` you get `1`. 

The idea of a multiplicative inverse extends to matrices, two matrices are inverses of each other if they multiply together to get the identity matrix.. The identity matrix for a `2times2` matrix is:
     `I_(n)=[(1, 0),(0, 1)]`

On page 69, Williams defines the properties of a matrix inverse by stating, "Let `A` be an `n times n` matrix. If a matrix `B` can be found such that `AB = BA=I_(n)`, then `A` is said to be invertible and `B` is called an inverse of `A`. If such a matrix does not exist, then `A`has no inverse."

Inverses of matrices relate to Gaussian elimination in that if you keep track of the row operations that you perform when reducing a matrix into the identity matrix, and you simultaneously perform the same operations to the identity matrix you end up with the inverse of the matrix you have reduced. This is particularly important to note because it extends to matrices of all different sizes since the identity matrix for an arbitrary `n times n` matrix always exists. 

However, there is also a formulaic way of producing the inverse of a `2times2` matrix, which we will present below.
     `A^(-1) = frac(1)(abs(A)) [(A_(22), -A_(12)), (-A_(21), A_(11))]`,
where `abs(A)` is `det A`.

Sources

 Williams, Gareth. Linear Algebra With Applications. Boston: Jones and Bartlett, 2011. Print. 

2x2 Matrix Calculators :

• To compute the Characteristic Polynomial of a 3x3 matrix,CLICK HERE.
• To compute the Trace of a 2x2 Matrix, CLICK HERE.
• To compute the Determinant of a 2x2 Matrix, CLICK HERE.
• To compute the Eigenvalues of a 2x2 Matrix, CLICK HERE.
• For the Classifying Equilibria of a 2x2 Matrix, CLICK HERE.
• To compute the Eigenvalues and Eigenvectors of a 2x2 Matrix, CLICK HERE.
• To multiply a 2x2 matrix by a scalar, CLICK HERE.
• To compute the Characteristic Polynomial of a 2x2 Matrix, CLICK HERE.