Linear Algebra True/False Final
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