What is what does it mean for a matrix to be invertible?

A matrix is said to be invertible (also called non-singular or non-degenerate) if there exists another matrix that, when multiplied with the original matrix, results in the identity matrix. In simpler terms, a matrix A is invertible if there is a matrix B such that:

A * B = B * A = I

Where I is the identity matrix. The matrix B is called the inverse of A, and it's denoted as A^-1.

Key points:

• Square Matrices: Only square matrices (matrices with the same number of rows and columns) can be invertible. A non-square matrix does not have an inverse.

• Determinant: A square matrix is invertible if and only if its determinant is non-zero. If the determinant is zero, the matrix is singular (non-invertible).

• Finding the Inverse: Various methods exist for finding the inverse of a matrix, including using Gaussian elimination (row reduction), adjugate matrices, or computational tools.

• Applications: Invertible matrices are essential in solving systems of linear equations, performing linear transformations, and various applications in mathematics, physics, engineering, and computer science. For instance, if you have a system of equations in matrix form, Ax = b, and A is invertible, then the solution is x = A^-1b.