Matrix Inverses

The inverse of an \(n \times n\) matrix \(M\) is another \(n \times n\) matrix, denoted \(M^{-1}\) such that \[ MM^{-1} = M^{-1}M = I \] where \(I\) is the \(n \times n\) identity matrix. Suppose a square matrix can be reduced to the identity by a sequence of elementary row operations. The inverse of this matrix is the result of applying the same operations, in the same order, to the identity.

Some important facts regarding matrix inverses include:

• Not all matrices are invertible. A square matrix is invertible if and only if its determinant is nonzero. Non square matrices do not have inverses, though they may have a right or left inverse.
• A matrix that is equal to its own inverse is called an involution.
• A matrix with integer entries will have an inverse with integer entries if and only if its determinant is -1 or 1.
• The determinant of a matrix and the determinant of its inverse are inversely proportional: \( \det(A^{-1}) = \frac{1}{\det(A)} \) for an invertible matrix \(A\).
• For an invertible matrix \(A\) and scalar \(c\), we have \( (cA)^{-1} = \frac{1}{c}A^{-1} \)
• For invertible matrices \(A\) and \(B\), we have \( (AB)^{-1} = B^{-1}A^{-1} \)
• The inverse of the transpose of a matrix is equal to the transpose of its inverse: \((A^T)^{-1} = (A^{-1})^T\) for an invertible matrix \(A\).