The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In other words, it is the number of dimensions in the output of a matrix transformation.
The rank of a matrix can be calculated by performing row reduction (Gaussian elimination) on the matrix and counting the number of non-zero rows or columns in the reduced matrix.
The rank of a matrix is important because it gives information about the dimension of the vector space spanned by the columns or rows of the matrix. It also determines whether a system of linear equations has a unique solution, no solution, or infinitely many solutions.
No, the rank of a matrix is a fundamental property of the matrix and does not change. However, the rank of a matrix can be affected by elementary row operations, such as multiplying a row by a non-zero scalar or swapping two rows.
The rank of a matrix is related to its determinant in that a matrix is invertible (has a non-zero determinant) if and only if its rank is equal to the number of rows or columns in the matrix. In other words, a matrix is invertible if and only if it has full rank.