If vector w is a linear combination of u and v, then u + v

If A is an n n matrix and x is a vector in Rn, then the product Ax is a linear combination of the columns of matrix A.

If vector u is a linear combination of vectors v and w , then we can write u = av + bw for some scalars a and b. 3.

. Matrix 120 001 000 is in reduced row-echelon form.

. A system of four linear equations in three unknowns is always inconsistent.

There exists a 3 4 matrix with rank 4.

If A is a 3 4 matrix and vector v is in R4, then vector Av is in R3.

If the 4 4 matrix A has rank 4, then any linear system with coefficient matrix A will have a unique solution.

There exists a system of three linear equations with three unknowns that has exactly three solutions.

There exists a 5 5 matrix A of rank 4 such that the system Ax = 0 has only the solution x = 0.

If matrix A is in reduced row-echelon form, then at least one of the entries in each column must be 1.

The system 123 456 000 x = 1 2 3 is inconsistent.

There exists a 2 2 matrix A such that A  1 2 =  3 4 .

If A is a nonzero matrix of the form  a b b a , then the rank of A must be 2

rank 111 123 136 = 3

. The system Ax = 0 0 0 1 is inconsistent for all 4 3

There exists a 2 2 matrix A such that A  1 1 =  1 2 and A  2 2 =  2 1 .

rank 222 222 222 = 2

11 13 15 17 19 21 1 3 1 = 13 19 21

There exists a matrix A such that A  1 2 = 3 5 7

Vector 1 2 3 is a linear combination of vectors 4 5 6 and 7 8 9

If the system Ax = b has a unique solution, then A must be a square matrix.

If A is any 4 3 matrix, then there exists a vector b in R4 such that the system Ax = b is inconsistent. 2

There exist scalars a and b such that matrix 0 1 a 1 0 b a b 0 has rank 3.

If v and w are vectors in R4, then v must be a linear combination of v and w . 25.

. If u, v, and w are nonzero vectors in R2, then w must be a linear combination of u and v. 26.

If v and w are vectors in R4, then the zero vector in R4 must be a linear combination of v and w . 27

If A and B are any two 3 3 matrices of rank 2, then A can be transformed into B by means of elementary row operations.

If vector u is a linear combination of vectors v and w , and v is a linear combination of vectors p, q, and r, then u must be a linear combination of p, q, r, and w . 29. A line

A linear system with fewer unknowns than equations must have infinitely many solutions or none.

The rank of any upper triangular matrix is the number of nonzero entries on its diagonal.

There exists a 4 3 matrix A of rank 3 such that A 1 2 3 =

The system Ax = b is inconsistent if (and only if) rref(A) contains a row of zeros.

If A is a 4 3 matrix of rank 3 and Av = Aw for two vectors v and w in R3, then vectors v and w must be equal. 34.

If A is a 4 4 matrix and the system Ax = 2 3 4 5 has a unique solution, then the system Ax = 0 has only the solution x = 0. 3

If vector u is a linear combination of vectors v and w , then w must be a linear combination of u and v. 36.

. If A =  u v w and rref(A) = 102 013 000 , then the equation w = 2u + 3v must hold. 37.

If A and B are matrices of the same size, then the formula rank(A + B) = rank(A) + rank(B) must hold.

If A and B are any two n n matrices of rank n, then A can be transformed into B by means of elementary row operations.

If a vector v in R4 is a linear combination of u and w , and if A is a 5 4 matrix, then Av must be a linear combination of Au and Aw . 40.

If matrix E is in reduced row-echelon form, and if we omit a row of E, then the remaining matrix must be in reduced row-echelon form as well.

. The linear system Ax = b is consistent if (and only if) rank(A) = rank A b . 4

If A is a 3 4 matrix of rank 3, then the system Ax = 1 2 3 must have infinitely many solutions.

If two matrices A and B have the same reduced rowechelon form, then the equations Ax = 0 and Bx = 0 must have the same solutions. 44

If matrix E is in reduced row-echelon form, and if we omit a column of E, then the remaining matrix must be in reduced row-echelon form as well.

If A and B are two 2 2 matrices such that the equations Ax = 0 and Bx = 0 have the same solutions, then rref(A) must be equal to rref(B). 46

A lower triangular 3 3 matrix has rank 3 if (and only if) the product of its diagonal entries is nonzero.

If adbc = 0, then the matrix  a b c d must have rank 2.

If vector w is a linear combination of u and v, then u + v + w must be a linear combination of u and u + v. 49. If

If the linear system Ax = b has a unique solution and the linear system Ax = c is consistent, then the linear system Ax = b + c must have a unique solution. 50. A

A matrix is called a 01-matrix if all of its entries are ones and zeros. True or false: The majority of the 01-matrices of size 3 3 have rank 3.