Indicate whether each of the following is true or false and

Indicate whether each of the following is true or false and explain your choice. If 3 is an eigenvalue of A, then 4 is an eigenvalue of the matrix B = A + 1(the identity matrix). If AX^rightarrow = B^rightarrow has a solution, then B^rightarrow is a vector in the row space of A. If A and B are square matrices and AB is invertible, then so are A and B.

Solution

a) The statement is TRUE

det (A - 3I) = 0 [given]

det(B - lambdaI) = det(A + I - \\lambdaI) = 0

Hence for det B its value must be 4

b) False, since B vector will belong to the column space of A and not necesarly row space of A

c) TRUE