MATH240 TRUE OR FALSE QUESTIONS

Question 1

If A is an augmented matrix of a system of linear equations, and there is a pivot in the last column
of the matrix, then the system is inconsistent.

Answer 1

True

Question 2

Every matrix that has a pivot in every row must be invertible.

Answer 2

False

Question 3

For any matrix A, A^T A is a square matrix. (sometimes, always, never

Answer 3

Always true

Question 4

For matrices B and C of equal size, the product BC exists.

Answer 4

Sometimes true

Question 5

If D is invertible, then D^2

is invertible.

Answer 5

Always true

Question 6

The eigenvalues of a matrix B are always the same as the eigenvalues of the reduced row echelon
form of B.

Answer 6

False

Question 7

Every orthogonal matrix C is orthogonally diagaonalizable.

Answer 7

False

Question 8

For any upper triangular matrix D, the eigenvalues are equal to the diagonal entries of the matrix.

Answer 8

True

Question 9

A 3 × 3 matrix has eigenvalues 5, 6, and 7. For the following questions, provide one of the answers: YES, NO,
or NOT ENOUGH INFORMATION. No justification is necessary.
i. (3pt) Is the matrix invertible?
ii. (3pt) Is the matrix diagonalizable?
iii. (3pt) Does the matrix A have exactly 3 eigenvectors?

Answer 9

i. Yes
ii. Yes
iii. No

Question 10

Every linear system of equations has the trivial solution in its solution set.

Answer 10

False

Question 11

There exists a linear system of equations with 2 equations in 2 variables which has exactly 2
solutions.

Answer 11

False

Question 12

If a linear system with matrix equation Ax = b has a unique solution, then A must be row
equivalent to the identity matrix.

Answer 12

False

Question 13

Any orthonormal set of vectors in R^3 not containing the zero vector is a basis for R^3

Answer 13

False

Question 14

If the columns of a matrix A form an orthogonal set where each vector is a unit vector, then A is
an orthogonal matrix.

Answer 14

False

Question 15

If a linear system has more variables than equations, it always has infinite solutions.

Answer 15

False

Question 16

If a matrix has a pivot in every column, then it must be invertible.

Answer 16

False

Question 17

For any matrix A, we have dim(col(A)) = dim(row(A)).

Answer 17

True

Question 18

The set of polynomials of degree at most 3 is a subspace of P4.

Answer 18

True

Question 19

The points (x, y) lying on the line y = 2x + 1 form a subspace of R
2

Answer 19

False

Question 20

The set of polynomials {x − 1,(x − 1)^2

,(x − 1)^3} span P3.

Answer 20

False

Question 21

If a linear transformation T is invertible, then it must be onto/surjective.

Answer 21

True

Question 22

If dim(V ) = 10 and V = span ({v1, v2, …v10}), then {v1, v2, …v10} is a basis for V .

Answer 22

True

Question 23

If v is an eigenvector for an eigenvalue λ, then the vector 2v must be an eigenvector for λ too.

Answer 23

True

Question 24

If a system of equations Cx = b has a unique solution, then the least squares solution IS the unique
solution.

Answer 24

True

Question 25

If U is an orthogonal matrix, then it must have columns which are orthonormal.

Answer 25

True

Question 26

Any matrix equation Mx = b always has at least one solution.

Answer 26

False

Question 27

(2pt) It is possible for a 4 x 4 matrix to have linearly independent columns that do not span R4.

Answer 27

False

Question 28

If U is an upper-triangular n x n matrix whose diagonal entries are all 1, then the columns of U span R".

Answer 28

True

Question 29

If a linear transformation T : R" → R" is onto, then it must be 1-1.

Answer 29

False

Question 30

The mapping T : R3 → R3 given by f(x, y, z) = (Ty, 2,0) is a linear transformation.

Answer 30

False

Question 31

If A is an invertible n x matrix, then the mapping T : R~ → ~ given by T(x) = Ax is a linear transformation that is both 1-1 and onto.

Answer 31

True

Question 32

If M and Q are similar matrices, then det (M) = det (Q).

Answer 32

True

Question 33

For any 5 x 5 matrix C', we have det (7C) = 7 det (C).

Answer 33

False

Question 34

A 2 × 2 matrix with real entries cannot have complex eigenvalues.

Answer 34

False

Question 35

A 2 x 2 matrix with real entries can have, as its eigenvalues, the two complex numbers 1 + 3i and 4 + 2i.

Answer 35

False

Question 36

The set S { (0,a,-1+b):a,b are R} a subspace of R^3

Answer 36

True

Question 37

If A is a 5 X 7 matrix, then the rank of A cannot be 6.

Answer 37

True

Question 38

The vector space In is isomorphic to In.

Answer 38

False