The inverse of a two-by-two matrix is another two-by-two matrix that, when multiplied by the original matrix, gives the identity matrix. Gauss-Jordan elimination and LU decomposition are two methods that can be employed to identify the inverse of a matrix.
A two-by-two matrix has two rows and two columns, with values a and b in the first row and c and d in the second row. Make a new matrix and place the values of d and negative b in the first row, then place the values of negative c and a in the second row. Multiply this new matrix by the value of 1 over the determinant, which is 1 over the quantity a times d minus b times c. The final result of this series of computations is the inverse of the original matrix.
Using software such as Mathematica simplifies the tedious process of calculating the inverse of a matrix, while also reducing the chance of making a computation error.