MATH 221 True/False Questions

FALSE. Matrix multiplication is not commutative.

(AB)C = (AC)B

FALSE. (AB)^T = (B^T)(A^T)

(AB)^T = (A^T)(B^T)

TRUE. The trivial solution is always a solution.

A homogeneous linear system is always consistent.

FALSE. det A^T = det A when A is (n x n).

det A^T = (-1) det A

FALSE. The n eigenvectors must be linearly independent.

A is diagonalizable if A has n eigenvectors.

FALSE. A unique linear system can have no solutions.

A linear system is unique if and only if it has exactly one solution.

TRUE. They are both the same as Ax = 0 for some x not equal to 0.

A matrix A is not invertible if and only if 0 is an eigenvalue of A.

FALSE. The product matrix is invertible, but the product of inverses should be in the REVERSE order.

A product of invertible (n x n) matrices is invertible, and the inverse of the product is the product of their inverses in the same order.

TRUE. A steady state vector has the property that Axx = x. In this case λ is 1.

A steady-state vector for a stochastic matrix is actually an eigenvector.

FALSE. The 2nd and 3rd statements are incorrect.

A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H (ii) u, v and u + v are in H (iii) c is a scalar and cu is in H.

TRUE. The eigenspace is the nullspace of A − λI.

An eigenspace of A is a null space of a certain matrix.

TRUE.

Any set B of 5 nonzero vectors in R^4 is linearly dependent.

FALSE. A line has to contain the origin to be a subspace.

Each line in R^n is a one-dimensional subspace of R^n.

FALSE. AB must be a (3 x 3) matrix, but this formula for AB implies that it is (3 x 1).

If A and B are (3 x 3) and B = [ b1 b2 b3 ], then AB = [ Ab1 + Ab2 + Ab3 ]

TRUE. The algorithm presented in this chapter tells us how to find the inverse in this case.

If A can be row reduced to the identity matrix, then A must be invertible.

TRUE.

If A is a (3 x 3) matrix, then det (2A) = 8 det A.

FALSE. The range could be a strict subset of R^m.

If A is an (m × n) matrix, then the range of the transformation x → Ax is R^m.

FALSE. The statement given is the converse of Theorem 6 (i.e., if A has n distinct eigenvalues, THEN A is diagonalizable.)

If A is diagonalizable, then A has n distinct eigenvalues.

FALSE. There exist matrices that are invertible but are not diagonalizable.

If A is invertible, then A is diagonalizable.

FALSE. The second half of the statement should be: "...also reduce the identity matrix to A^(−1).

If A is invertible, then elementary row operations that reduce A to the identity matrix also reduce A^(-1) to the identity matrix.

TRUE.

If A is invertible, then the inverse of A^(-1) is A itself.

TRUE. Its inverse is B^(-1) B^(-1) A^(-1).

If A, B are invertible (n × n) matrices, then AB^2 is invertible.

TRUE. This follows from AP = PD and formulas in the proof of the Diagonalization Theorem.

If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A.

FALSE. The vector in Ax = λx must be NONZERO.

If Ax = λx for some scalar λ, then x is an eigenvector of A.

FALSE. The converse is true (i.e., if two rows or two columns are the same, or a row or a column is zero, THEN the det A is zero.)

If det A = 0, then two rows or two columns are proportional, or a row or a column is zero.

FALSE. The converse is true (i.e., if two rows or two columns are the same, or a row or a column is zero, THEN the det A is zero.)

If det A = 0, then two rows or two columns are the same, or a row or a column is zero.

FALSE.

If the characteristic polynomial of a (4 × 4) matrix A is (λ − 1)^2(λ − 2)(λ − 3), then the eigenspace of A corresponding to eigenvalue 1 has dimension 2.

FALSE.

If the characteristic polynomial of a 5 × 5 matrix A is (λ + 2)^2(λ − 2)^3, then the eigenspace of A corresponding to eigenvalue −2 has dimension 2.

TRUE. dim Col A = 6 - 2 = 4.

If the null space of a 5 × 6 matrix A is 2 dimensional, then the column space of A is 4 dimensional.

TRUE. Both changes multiply the determinant by -1 and -1*-1=1.

If two row changes are made in succession, then the new determinant equals the old determinant.

FALSE. The converse is true (i.e., if v1, . . ., vr are eigenvectors that correspond to distinct eigenvalues λ1, . . ., λr of an (n x n) matrix A, then the set {v1, . . ., vr} is linearly independent.)

If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues.

FALSE. The reduced echelon form is unique.

In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.

FALSE. An invertible matrix has to be a square matrix.

Let A be a (4 x 5) matrix. If Ax = 0 only has the trivial solution x = 0, then A is invertible.

FALSE. The solution set of Ax = b could be empty.

Let A be an m × n matrix and b ∈ R^m be a nonzero vector. If the solution set of Ax = 0 is infinite, then the solution set of Ax = b is also infinite.

FALSE. We have (A + B)^2 = A^2 + B^2 + AB + BA, but in general, AB does not equal BA.

Let A, B denote square matrices. Then (A + B)^2 = A^2 + 2AB + B^2.

FALSE. The (i, j)-cofactor of a matrix A is: (−1)^(i+j) det Aij.

The (i, j)-cofactor of a matrix A is the matrix Aij obtained by deleting the ith row and jth column from A.

FALSE. We can expand down any row or column and get same determinant.

The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row.

TRUE. There are more columns than entries in each column, thus by Theorem 8, they are linearly dependent.

The columns of any 4 x 5 matrix are linearly dependent.

FALSE. This is only true when A is triangular.

The determinant of A is the product of the diagonal entries in A.

FALSE. It is the product of the diagonal entries.

The determinant of a triangular matrix is the sum of the entries on the main diagonal.

FALSE. This is only true for triangular matrices.

The eigenvalues of a matrix are on its main diagonal.

FALSE. The pivot position cannot be in the augmentation column.

The equation Ax = b is consistent if the augmented matrix [ A b ] has a pivot position in every row.

FALSE. It's in R^n.

The null space of an m × n matrix is in R^m.

TRUE.

The second row of AB is the second row of A multiplied on the right by B.

TRUE.

The transpose of a sum of matrices equals the sum of their transposes.