LINEAR ALGEBRA FINAL TEST TRUE/FALSE QUESTIONS WITH CORRECT ANSWERS

LINEAR ALGEBRA FINAL TEST
TRUE/FALSE QUESTIONS WITH
CORRECT ANSWERS
[0 0 0 1] is matrix in rref - Answer-True

If the only solution of Ax = 0 is the zero solution, then the rows of A are linearly
independent - Answer-False, A is could be inconsistent (all zero bottom row)

If the matrix A has m rows and n columns, then the matrix A(A^T)A will also have m
rows and n columns - Answer-True

If A is a 7x4 matrix, then the nullity of (A^T) cannot be less than 3 - Answer-True

If two matrix vector products Au and Av are equal, then the vectors u and v must be
equal to each other - Answer-False

An nxn elementary matrix will never have more than n+1 nonzero entries - Answer-True

A set consisting of two parallel vectors must be linearly dependent - Answer-True

If AB is defined, then (AB)^-1 = (A^-1)(B^-1) - Answer-False

If A is invertible then the system of equations Ax = b has a unique solution for every b -
Answer-True because if A is invertible, it is also linearly independent

If span({v1, v2}) = span({v1, v2, v3}), then v3 must be a linear combination of v1 and v2
- Answer-True(End of M1)

If det A = 0, then A is not invertible - Answer-True, det must be > 0 to be invertible

If lambda is an eigenvalue of an invertible matrix A, then lambda cannot equal 0 -
Answer-True

If A is an mxn matrix then dim(Null(A)) + dimRow(A) = n - Answer-True, nullity =
columns(n) - rank(non-zero rows)

Every n x n diagonal matrix is diagonalizable - Answer-True (want explanation)

For any matrix A, the dimension of row space of A and the dimension of the row space
of A^T are equal - Answer-True

If A and B are any two n x n matrices then det(A+B) = det(A) + det(B) - Answer-False