An invertible matrix, also known as a nonsingular matrix, is a square matrix whose determinant is not equal to zero. This means that the matrix has an inverse, which is another matrix that, when multiplied by the original matrix, results in the identity matrix.
A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the other vectors. In the context of matrices, this means that none of the columns can be expressed as a linear combination of the other columns.
No, an invertible matrix is not always linearly independent. A matrix can be invertible but still have linearly dependent columns, meaning that one or more columns can be expressed as a linear combination of the other columns.
An invertible matrix must have linearly independent columns, but the reverse is not necessarily true. This means that if a matrix is invertible, it must also be linearly independent, but a matrix can be linearly independent without being invertible.
Yes, there are exceptions. For example, a square matrix with only one non-zero column will be linearly independent but not invertible, as its determinant will be equal to zero. Similarly, a square matrix with only one non-zero row will be linearly independent but not invertible.