Inverting a 2 x 2 matrix

• Begin with a 2 x 2 matrix in the form [a b; c d].

• Remember the label of the elements will be reversed when inverted. The element in the top left corner, which is labeled ‘a’, and bottom right corner, which is labeled ‘d’ will swap in an inverse matrix.

• Similarly, the elements in the opposite corners (top right, labeled ‘b’, and bottom left, labeled ‘c’) will also swap, but these will become negative.

• This means the inverse matrix is in the form [d -b; -c a].

• However, these elements all divide by the determinant of the initial matrix.

• Determine the determinant of the original matrix. In a 2 x 2 matrix, the determinant is given by the product of the elements in the leading diagonal (ad) minus the product of the elements in the other diagonal (bc).

• This will look like: Det = (ad - bc).

• Now, divide each element in [d -b; -c a] by the determinant (ad - bc). The resulting matrix is the inverse.

• Hence, the inverse of a 2 x 2 matrix [a b; c d], given it is non-singular, is [d/(ad-bc) -b/(ad-bc); -c/(ad-bc) a/(ad-bc)].

• If the determinant of the matrix is zero, then it is not invertible as we cannot divide by zero. Such matrices are called “singular”.

• An invertible matrix or a non-singular matrix is also referred to as full-rank.

• Be aware of the properties of matrix inverses, including: (A^-1)^-1 = A, (kA)^-1 = 1/k(A^-1) for any nonzero scalar k, and (AB)^-1 = B^-1A^-1 (the law of reversal).

• Remember to check the inverse matrix calculated by multiplying it with the original matrix. The resultant matrix should be the identity matrix [1 0; 0 1], confirming that the calculations are correct.