invertible matrix

invertible matrix, a square matrix such that the product of the matrix and its inverse generates the identity matrix. That is, a matrix M, a general n × n matrix, is invertible if, and only if, M ∙ M⁻¹ = I_n, where M⁻¹ is the inverse of M and I_n is the n × n identity matrix. Often, an invertible matrix is referred to as a nonsingular (or nondegenerate) matrix.

The identity matrix is a square matrix with values of 1 along the main diagonal (starting in the upper left corner of the matrix and ending in the bottom right corner) and zeros in all other locations. As an example, the following is the 4 × 4 identity matrix: .

Finding the inverse of a matrix is referred to as matrix inversion. This process takes a matrix from its original form to its inverse form through operations involving the identity matrix. In this process, certain conditions must be true. First, the original matrix must be a square matrix, meaning that there is the same number of columns as rows. Rectangular matrices, where the number of rows and number of columns differ, do not have multiplicative inverses. Most importantly, a matrix is invertible if, and only if, the determinant of the matrix is not zero. Therefore, any square matrix that has a complete column or a complete row that is only zeros cannot be an invertible matrix, since the identity matrix requires one value of 1 in a column or in a row, which cannot be obtained when a full column or a full row contains only zeros. This also means that the zero matrix is not an invertible matrix.

All identity matrices are invertible, since the determinant of all identity matrices is 1, which is a nonzero value. The inverse of an identity matrix is the same identity matrix. Thus, when an identity matrix is multiplied by its inverse (which is the same identity matrix), the result is the same identity matrix. Any matrix that is its own inverse is called an involutory matrix (a term that derives from the term involution, meaning any function that is its own inverse).

Invertible matrices have the following properties:

• 1. If M is invertible, then M⁻¹ is also invertible, and (M⁻¹)⁻¹ = M.
• 2. If M and N are invertible matrices, then MN is invertible and (MN)⁻¹ = M⁻¹N⁻¹.
• 3. If M is invertible, then its transpose M^T (that is, the rows and columns of the matrix are switched) has the property (M^T)⁻¹ = (M⁻¹)^T. That is, the inverse of the transpose of M is equal to the transpose of the inverse of M.