vela said:
An invertible matrix, also known as a nonsingular matrix, is a square matrix that has a unique inverse matrix. This means that when multiplied together, the inverse matrix and the original matrix result in the identity matrix (a square matrix with 1's along the main diagonal and 0's everywhere else).
A matrix is invertible if its determinant is non-zero. This can be determined by using the determinant formula for a square matrix, where the determinant is calculated by taking the sum of the products of the elements in each row and column. If the determinant is equal to zero, the matrix is not invertible.
No, not all matrices can be inverted. Only square matrices can be inverted, and even then, only matrices with a non-zero determinant can be inverted. Matrices that do not have a unique inverse are called singular matrices.
Inverting a matrix is useful for solving systems of linear equations, as well as for calculating the inverse of a linear transformation. It also allows for easier computation and manipulation of matrices in certain mathematical operations.
The inverse of a matrix can be found by using the inverse matrix formula, which involves finding the adjugate (or adjoint) matrix of the original matrix and then dividing it by the determinant of the original matrix. Another method is by using elementary row operations to reduce the original matrix to the identity matrix, and performing the same operations on the identity matrix to obtain the inverse.