An n × n matrix A is said to be invertible (or nonsingular) if there is another...

An n × n matrix A is said to be invertible (or nonsingular) if there is another n × n matrix B with the property that

AB = BA = I_n

where I _n denotes the n × n identity matrix. (See Exercise 20.) The matrix B is called an inverse to the matrix A. Exercises 30–38 concern various aspects of matrices and their inverses.

If A is a 3 × 3 matrix and det A 0, then there is a (somewhat complicated) formula for A⁻¹. In particular,

where A_{i j} denotes the submatrix of A obtained by deleting the ith row and j th column (see Definition 6.5). Use this formula to find the inverse of

More generally, if A is any n × n matrix and det A 0, then

where adj A is the adjoint matrix of A, that is, the matrix whose i jth entry is (−1)^i+j |A_ji|. (Note: The formula for the inverse matrix using the adjoint is typically more of theoretical than practical interest, as there are more efficient computational methods to determine the inverse, when it exists.)