The multiplicative inverse of a matrix is a matrix that when multiplied by the original matrix from both sides yields the identity matrix. In other words, the multiplicative inverse of a matrix \(A\) is a matrix \(B\) such that $$AB = BA = I,$$ where \(I\) is the identity matrix. A matrix \(A\) is invertible if it has an inverse. An invertible matrix is called a nonsingular matrix in some references. In the post on matrix operations, we introduced the inverse of a matrix, the inversion unary operation, and invertible matrices briefly. In this post, we will discuss the inverse of a matrix and invertible matrices in detail.

It is easy to see that if a matrix \(A\) is invertible, its inverse is unique. The inverse of a matrix \(A\) is denoted by \(A^{-1}.\)

Remark. A square matrix is singular if it is not invertible.

#### Basic properties of the inverse of a matrix

The proof of the following result is straightforward:

**Result**. Let \(A\) be an \(n \times n\) invertible matrix and \(b\) a vector in \(\mathbb{R}^n\). The matrix equation \(Ax = b\) has the unique solution \(x = A^{-1}b\).

Now, we proceed to explain some of the basic properties of invertible matrices:

- If a matrix is invertible, then its inverse is also invertible. Moreover, the inverse of the inverse of a matrix is the same as the original matrix. In other words, if \(A\) is invertible, then $$(A^{-1})^{-1} = A.$$
- The multiplication of a finite number of invertible matrices is invertible. Furthermore, the inverse of the multiplication of invertible matrices is the multiplication of their inverses in the reverse order. In other words, if \(A_i\) is invertible for each \(i\), then $$(A_1 \cdots A_n)^{-1} = A^{-1}_n \cdots A^{-1}_1.$$

Technical comment. The set of all \(n \times n\) real matrices equipped with the matrix multiplication is a monoid and its absorbing element of the matrix zero. The set of all invertible \(n \times n\) real matrices is a subgroup of the monoid of all \(n \times n\) real matrices.

#### Characterizations of invertible matrices

This result gives a number of criteria for the invertibility of real matrices:

**Result**. Let \(A\) be an \(n \times n\) real matrix. Then, the following statements are equivalent:

- \(A\) is invertible.
- \(A\) is row equivalent to the identity matrix \(I_n\).
- The matrix equation \(Ax = 0\) implies that \(x\) is the \(n\)-dimensional zero vector. In other words, the matrix equation \(Ax = 0\) has only the trivial solution.
- The columns of \(A\) are linearly independent.
- The linear map \(x \mapsto Ax\) is 1-to-1.
- The matrix equation \(Ax = b\) has at least one solution for each \(n\)-dimensional vector \(b\).
- The columns of \(A\) span (generated) the \(n\) dimensional real vector space \(\mathbb{R}^n\).
- The linear map \(x \mapsto Ax\) is onto.
- There is a square matrix \(B\) such that \(BA = I\).
- There is a square matrix \(B\) such that \(AB = I\).
- The transpose of the matrix \(A\) is invertible.