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Given that A and B are n × n matrices and T is a linear transformation....

Given that A and B are n × n matrices and T is a linear transformation. Determine which of the following is FALSE.

(a) If AB is not invertible, then either A or B is not invertible.

(b) If Au = Av and u and v are 2 distinct vectors, then A is not invertible.

(c) If A or B is not invertible, then AB is not invertible.

(d) If T is invertible and T(u) = T(v), then u = v.

(e) none of these

Given An×nx = 0 has only the trivial solution, determine which of the following is FALSE.

(a) Ax = 0 does not have any free variable and thus every column of A is a pivot column.

(b) Ax = b has exactly one solution A−1 b for every b ∈ Rn

(c) A is row equivalent to In.

(d) Every row of A has a pivot position.

(e) none of these

Given that columns of Bn×n are linearly dependent, determine which of the following is FALSE.

(a) Bx = 0 has nontrivial solutions and B is not invertible.

(b) Columns of Bn×n do not span Rn .

(c) Bx = b is inconsistent for some b.

(d) BT x = 0 has more than one solution.

(e) None of these

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