Linear Algebra Chapter 2-3.2 True/False

Suppose A is an n×n matrix. If the linear transformation TA(x)=Ax is one-to-one, then the columns of A form a linearly dependent set.

no

For any matrix A, we have the equality 2A+3A=5A.

True

If A is an m×n matrix then A^TA and AA^T are both defined.

True

Let V be the subset of ℝ^3 consisting of the vectors [a,b,c] with abc=0. V contains the zero vector.

True

Assume A is an n×n matrix A row replacement operation does not affect the determinant of a matrix.

True

Assume A is an n×n matrix If two columns of a matrix are the same, then the determinant of that matrix is zero

True

Suppose A is an n×n matrix If A is invertible, then the equation Ax=b has exactly one solution for all b in ℝ^n

Yes

Assume A is an n×n matrix. The determinant of a triangular matrix is the sum of the entries of the main diagonal.

false

Suppose A is an n×n matrix. If the equation Ax=0 has the trivial solution, then the columns of A span ℝ^n.

maybe

Suppose A is an 5×5 matrix. If A has three pivots, then ColA is a two-dimensional plane.

no

Assume A is an n×n matrix. A determinant of an n×n matrix can be defined as a sum of determinants of (n−1)×(n−1) submatrices.

true

Suppose A is an 5×5 matrix. If Ax=0 has only the trivial solution, then ColA=ℝ^5

yes

Assume A is an n×n matrix If det A is zero, then two columns of A must be the same, or all of the elements in a row or column of A are zero.

False

Assume A is an n×n matrix If the columns of A are linearly independent, then det A=0.

False

Assume A is an n×n matrix det (A+B) = det A + det B

False

For any matrices A and B, if the product AB is defined, then BA is also defined.

False

If A is a 5×4 matrix, and B is a 4×3 matrix, then the entry of AB in the 3rd row / 4th column is obtained by multiplying the 3rd column of A by the 4th row of B

False

If B is an echelon form of a matrix A, then the pivot columns of B form a basis for the column space of A.

False

Let V be the subset of ℝ^3 consisting of the vectors [a,b,c] with abc=0. V is a subspace of ℝ^3

False

Let V be the subset of ℝ^3 consisting of the vectors [a,b,c] with abc=0. V is closed under vector addition, meaning that if u and v are in V then u+v is in V.

False

Suppose A and B are invertible matrices. (AB)^−1=A^−1B^−1.

False

Suppose A and B are invertible matrices.(A+B)^2=A^2+B^2+2AB.

False

The null space of an m×n matrix is a subspace of ℝ^m

False

Suppose A is an n×n matrix A square matrix with two identical columns can be invertible.

No

For any matrix A, there exists a matrix B so that A+B=0.

True

Let V be the subset of ℝ^3 consisting of the vectors [a,b,c] with abc=0. V is closed under scalar multiplication, meaning that if u is in V and c is a real number thenthen cu is in V.

True

Suppose A and B are invertible matrices. (In−A)(In+A)=In−A^2.

True

Suppose A and B are invertible matrices. A+B is invertible.

True

Suppose A and B are invertible matrices. A^7 is invertible.

True

The column space of an m×n matrix is a subspace of ℝ^m

True

The columns of an invertible n×n matrix form a basis for ℝ^n

True

The columns of an invertible n×n matrix form a basis for ℝ^n.

True

The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of ℝ^n

True

Suppose A is an n×n matrix If A^T is row equivalent to the n×n identity matrix, then the columns of A span ℝn.

Yes

Suppose A is an n×n matrix The product of any two invertible matrices is invertible.

Yes

Suppose A is an n×n matrix. If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot points.

Yes

Suppose A is an n×n matrix. If the linear transformation TA(x)=Ax is onto, then it is also one-to-one.

Yes

Suppose A is an n×n matrix. If the transpose of A is not invertible, then A is also not invertible.

Yes

Suppose A is an 5×5 matrix. If A has two pivots, then the dimension of NulA is 2.

no

Suppose A is an 5×5 matrix. If rankA=4 , then the columns of A form a basis of ℝ^5.

no

Assume A is an n×n matrix. The (i,j) minor of a matrix A is the matrix Aij obtained by deleting row i and column j from A.

true

Assume A is an n×n matrix. The absolute value of the determinant of A equals the volume of the parallelepiped determined by the columns of A

true

Assume A is an n×n matrix. The cofactor expansion of det A along the first row of A is equal to the cofactor expansion of det A along any other row.

true