4. Group up the numbers that are equal if A is an m × n matrix which is also the

Question

4. Group up the numbers that are equal if A is an m × n matrix which is also the coefficient matrix of a linear system.

(a) Number of rows of the matrix A

(b) Number of columns of the matrix A

(c) Determinant of the matrix A (if it exists)

(d) Rank of the matrix A

(e) Nullity of the matrix A

(f) Number of non zero rows in the row echelon form of the matrix A

(g) Number of non zero columns in the row echelon form of the matrix A

(h) Number of leading variables of the system

(k) Number of free variables of the system

(l) Number of leading variables of the system

(m) m

(n) n

(o) Dimension of the rowspace of A

(p) Dimension of the columnspace of A

(q) Dimension of the solution space N(A)

Explanation / Answer

Assuming m < n and rank (A) = m

1) Number of rows of the matrix A = Rank of the matrix A = Number of non zero rows in the row echelon form of the matrix A = Number of leading variables of the system = Dimension of the rowspace of A = Dimension of the columnspace of A = m

2) Number of columns of the matrix A = Number of non zero columns in the row echelon form of the matrix A = n

3) Nullity of the matrix A = Dimension of the solution space N(A) = Number of free variables of the system = n-m

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