Frequently Asked Questions
What is matrix rank?
The maximum number of linearly independent rows or columns. Equivalently, the dimension of the image of the linear map defined by the matrix.
Rank versus determinant?
For a square matrix, full rank is equivalent to a nonzero determinant. Rank generalizes to non-square matrices and is more numerically stable to compute via row reduction or SVD.
What is rank-nullity?
For an m by n matrix, rank + nullity = n where nullity is the dimension of the null space (vectors x with A times x equal to zero). It is the fundamental relation between image and kernel dimensions.
How do you find the rank of a matrix by row reduction?
Reduce the matrix to row echelon form and count the nonzero (pivot) rows. That count is the rank, equal to the number of linearly independent rows or columns.
What does full rank mean?
A matrix has full rank when its rank equals the smaller of its row and column counts. A square matrix of full rank is invertible; less than full rank means the rows or columns are linearly dependent and the determinant is zero.
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