Mark each statement True or False. a: If an m × n matrix A has a pivot position

Question

Mark each statement True or False.

a: If an m × n matrix A has a pivot position in every row, then the equation Ax = b has a unique solution for each b in R m.

b: If A is a 6 × 5 matrix, the linear transformation x 7 Ax cannot map R 5 onto R 6 .

c: If A is 3 × 3 matrix, then det (3A) = 3 detA.

d: If AB = 0, then either A = 0 or B = 0.

e: If A and B are n × n matrices, with detA = 2 and detB = 3, then det(A + B) = 5.

f: If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets.

Explanation / Answer

a). False

The system is inconsistent if [A b] has a pivot in the last ("b") column. The system is consistent if the matrix A has a pivot in every row.

B). False

c).False.

There is a property that says that, for a constant c in an n x n matrix:
det(c*A) = c^n * det(A)

so det (3A) = 3 ^3detA

d). False.

e).False

det(A+B) is not equal to det(A) +det(B)

but det(AB) = det(A) . det(B)

f). True

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