False

If x is a nontrivial solution of Ax=0, then every entry in x is nonzero.

Is the statement true or false?

Is the statement true or false?

True

The equation Ax = b is homogeneous if the zero vector is a solution.

Is the statement true or false?

True

The equation x = x2u + x3v with x2 and x3 free (and neither u nor v a multiple of the other), describes a plane through the origin.

False

The solution set of Ax = b is always obtained by translating the solution set of Ax = 0.

False

If A is 3×3 with three pivot positions, then Ax = 0 has a nontrivial solution.

True

If A is 3×3 with three pivot positions, then Ax = b has a solution for every b in R3.

True

If A is 3×3 with two pivot positions, then Ax = 0 has a nontrivial solution.

False

If A is 3×3 with two pivot positions, then Ax = b has a solution for every b in R3.

True

If A is 3×2 with two pivot positions, then Ax = 0 has a nontrivial solution.

False

If A is 3×2 with two pivot positions, then Ax = b has a solution for every b in R3.

True

If A is 2×4 with two pivot positions, then Ax = 0 has a nontrivial solution.

True

If A is 2×4 with two pivot positions, then Ax = b has a solution for every b in R2.

True

Two vectors in R3 are linearly dependent if and only if they both lie on same line through the origin.

True

If a matrix has more rows than columns, then its rows must be linearly dependent.

True

If vectors x, y, and z are such that z belongs to span{x,y}, then the collection {x,y,z} must be linearly dependent.

False

If the columns of a matrix are linearly dependent, then the matrix must have more columns than rows.

True

If a collection of vectors in Rn contains the zero vector, then the collection must be linearly dependent.

False

If the third column of a 4×4 matrix is not a linear combination of the other three columns, then the columns of this matrix must be linearly independent.

True

If the set of vectors {v1, v2, v3, v4, v5} is linearly independent, then so is {v1, v2, v4, v5}.

True

Elementary row operations on an augmented matrix never change the

solution set of the associated linear system

solution set of the associated linear system

False

Two matrices are row equivalent if they have the same number of rows.

False

Let A be an mxn matrix, and assume m > n. Additionally, assume that the

matrix A has n pivots. Then for any vector b ∈ Rm, the equation Ax = b is consistent.

matrix A has n pivots. Then for any vector b ∈ Rm, the equation Ax = b is consistent.

True.

If for some matrix A, and some vectors x,b, we have Ax = b, then b is a

linear combination of the column vectors of A.

linear combination of the column vectors of A.

True

If A and B are both invertible nxn matrices, then AB is invertible.

True

Let A and B be nxn matrices. Assume that AB = In. Then, BA = In.

The homogeneous equation Ax = 0 is always consistent. this is

because after row reducing you will never have a row of all zeros and a nonzero in the

augmented part, (because the augmented part is all zeros). In fact the zero vector is always

a solution to a homogeneous equation.

because after row reducing you will never have a row of all zeros and a nonzero in the

augmented part, (because the augmented part is all zeros). In fact the zero vector is always

a solution to a homogeneous equation.

Let A be an mxn matrix. Then, the homogeneous equation Ax = 0 is consistent

if and only if the augmented matrix [ A | 0 ] has a pivot in every row.

if and only if the augmented matrix [ A | 0 ] has a pivot in every row.

True.

Let A be an mxn matrix and let A^T be its transpose. Then A^T A is an nxn

matrix so its determinant is well defined. In fact, the determinant det(A^T A) is greater than

or equal to zero.

matrix so its determinant is well defined. In fact, the determinant det(A^T A) is greater than

or equal to zero.

False

The equation Ax = b has at least one solution

for each b ∈ R

n.

for each b ∈ R

n.

True

If the equation Ax = 0 has nontrivial solutions

then A has fewer than n pivot positions.

then A has fewer than n pivot positions.

True

If AT

is not invertible then A is not invertible

is not invertible then A is not invertible

True

If there is an n × n matrix D such that

AD = I then there is also an n × n matrix

C such that CA = I.

AD = I then there is also an n × n matrix

C such that CA = I.

True

If the columns of A are linearly independent

then they span Rn

then they span Rn

True

If the equation Ax = b has at least one

solution for each b ∈ R

n, then the solution

is unique for each b.

solution for each b ∈ R

n, then the solution

is unique for each b.

True

If the linear transformation x → Ax maps

Rn onto Rn, then A has n pivot positions

Rn onto Rn, then A has n pivot positions

True

If there is b ∈ R

n such that the equation

Ax = b is inconsistent, then the transformation

x→ Ax is not one-to-one

n such that the equation

Ax = b is inconsistent, then the transformation

x→ Ax is not one-to-one

True

Row operations do not affect linear dependence

relations among the columns of

a matrix. 1

relations among the columns of

a matrix. 1

False

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

True

If a matrix A is invertible, then the inverse of A^−1 is A itself.

False

If A = [a b; c d]
and ad=bc, then A is invertible.

True

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

False

If a matrix A is invertible, then the same elementary row operations that reduce A to the identity I also reduce A^−1 to I.

True

If there is an n×n matrix D such that AD=I, then DA=I.

Note: The matrix A is n×n.

True

If the columns of an n×n matrix A are linearly independent, then the columns of A span Rn

True

If the equation Ax=b has at least one solution for each b in Rn, then the solution is unique for each b.

Note: The matrix A is n×n.

False

If the linear transformation x ->Ax maps Rn to Rn, then A always has n pivot positions.

Note: The matrix A is n×n.

True

If there is a b in Rn such that the equation Ax=b is inconsistent (i.e. has no solution), then the transformation x->Ax is not one-to-one.

Note: The matrix A is n×n.