The Rank of a Matrix
Searle (1982)..."Xa = 0 being true for some a unequals 0 means the columns (vectors) in X (matrix) are linearly dependent, whereas it being true only for a (vector) = 0 means they are linearly independent."
When the columns (or rows) of a square matrix are linearly dependent, that matrix has no inverse. In other words, it is singular. A zero determinant means the vectors composing of a square matrix are linearly dependent.
The rank of a matrix is the number of linearly independent rows (and columns) in the matrix.
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