For a is m× m and b is n×k. Write AB=[AbAb2……Ab], where bi are the ith column of B. That is, each column of AB can be expressed as a linear combination of the column of A, so the number of linearly independent columns in AB can not be more than the number of linearly independent columns in A. Thus, rank(AB)< rank(A). Similarly, each row of AB can be expressed as a linear combination of the rows of B from which we get rank(AB)< rank(B) Corolla 1. For any matrix A rank(A)=rank(A'A)=rank(AA,) 2. Let a is mx n matrix. b is m x m matrix and c is n x n matrix. Then if B and c are nonsingular matrices. it follows that rank(BAC)=rank(BA)=rank(AC)=rank(A) Set B-I, if A is M X n and C is a square matrix of rank n, then rank(AC)=rank(A) 3.6 Determinant For a 2 x 2 matrix A, the area of the parallelogram formed by its columns is the determinant of A, denoted as A Propositio The determinant of a matrix is nonzero if and only if it has full rank Useful results CD=CIDI, for a constant c, and k x k matric D CD =C| DI for two matri.z C and DProof: For A is m × n and B is n × k. Write AB = [Ab1 Ab2..... Abk], where bi are the ith column of B. That is, each column of AB can be expressed as a linear combination of the column of A, so the number of linearly independent columns in AB can not be more than the number of linearly independent columns in A. Thus, rank(AB) ≤ rank(A). Similarly, each row of AB can be expressed as a linear combination of the rows of B from which we get rank(AB) ≤ rank(B). Corollary: 1. For any matrix A, rank(A)=rank(A’A)=rank(AA’). 2. Let A is m × n matrix, B is m × m matrix, and C is n × n matrix. Then if B and C are nonsingular matrices, it follows that rank(BAC)=rank(BA)=rank(AC)=rank(A) Example: Set B=I, if A is M × n and C is a square matrix of rank n, then rank(AC)=rank(A). 3.6 Determinant For a 2 × 2 matrix A, the area of the parallelogram formed by its columns is the determinant of A, denoted as |A|. Proposition: The determinant of a matrix is nonzero if and only if it has full rank. Useful results: 1. |cD| = c k |D|, for a constant c, and k × k matrix D. 2. |CD| = |C| · |D| for two matrix C and D. 13