Linear Algebra True/False Final

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Matrix[1/3 -1/3;1/3 1/3] represents a rotation

False

det(4A)=4det(A) of all 2x2 matrices A

False

If A and B are nxn matrices, and vector x is in the kernal of both A and B, then x must be in the kernal of matrix AB as well

True

If A is a 3x3 matrix and the system AX=[1;2;3] has a unique solution, then the system Ax=0 has only the solution x=0

True

If A is a 3x3 matrix and vecotr x is in R4, then Ax is in R3

True

If A is any invertible matrix, then A commutes with A-1

True

If A is invertible, then 0 failes to be an eigenvalue of A

True

If an nxn matrix A is diagonalizable, then there must be a basis of Rn consisting of eigenvectors of A

True

If matrix A is diagonalizable, then its transpose AT must be diagonalizable as well

True

If matrix A is invertible, then 3A must be invertible.

True

If matrix A is symmetric and matrix B is orthogonal, then matrix B-1AB must be symmetric

True

If the 5x5 matrix A has rank 5, then any linear system with coefficient matrix A will have a unique solution

True

Matrix [4 5;-5 4] represents a rotation combined with a scaling

True

The determinate of any diagonal nxn matrix is the product of its diagonal entries

True

The eigenvalues of any triangular matrix are its diagonal entries

True

If 2 is an eigenvalue of an nxn matrix A, then 8 must be an eigenvalue of matrix A3

True

If 2u+3v+5w=0, then the vectors u, v, w must be linearly dependent

True

If an invertible matrix A is diagonalizable, then A-1 is diagonalizable as well

True

All diagonalizable matrices are invertible

False

If A and B are matrices of the same size, then the formula rank(A-B)=rank(A)-rank(B)

False

If A is a 2x4 matrix and B is a 2x5 matrix, then AB will be a 5x2 matrix

False

If ATA=AAT for any nxn matrix A, then A must be orthogonal

False

If a matrix is diagonaliazble, then the algebraic multiplicity of each its eigenvalues must equal the geometric multiplicity

False

Let A be a 4x5 matrix, then Ker(A) is a subspace of R4

False

Let V be a subspace of Rn, then the intersection of V and V(perpendicular) has a non-zero vector

False

The equation (AB)T=ATBT holds for all nxn matrices A and B

False

The image of 3x5 matric is a subspace of R5

False

The kernal of any invertible matrix consists of the zero vector only

False

There exists an invertible matrix with two identival rows

False

ATA is symmetric for all matricies A

True


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