Linear Algebra True/False Chapter 1
A matrix with a row or column of zeros can have an inverse.
False
Every matrix has a unique REF.
False
A homogeneous linear system in n unknowns whose corresponding augmented matrix has a RREF with r leading 1's has n-r free variables.
True
An expression of an invertible matrix A as a product of elementary matrices is unique.
False
Every matrix has a unique row echelon form.
False
For all square matrices A and B of the same size, it is true that (A+B)²=A²+2AB+B².
False
For all square matrices A and B of the same size, it is true that A²-B²=(A-B)(A+B).
False
If A and B are (2x2) matrices, then AB = BA.
False
If A and B are invertible matrices of the same size, then AB is invertible and (AB)-¹=A-¹B-¹
False
If A, B, and C are square matrices of the same order such that AC = BC, then A = B.
False
If a linear system has more unknowns than equations, then it must have infinitely many solutions.
False
If an elementary row operation is applied to a matrix that is in REF, the resulting matrix will still be in REF.
False
If each equation in a consistent linear system is multiplied through by a constant c, then all solutions to the new system can be obtained by multiplying solutions from the original system by c.
False
If every column of a matrix in REF has a leading 1, then all entries that are not leading 1's are zero.
False
If the RREF of the augmented matrix for a linear system has a row of zeros, then the system must have infinitely many solutions.
False
If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent.
False
Multiplying a row of an augmented matrix through by zero is an acceptable elementary row operation.
False
The linear system with corresponding augmented matrix [2 -1 4 0 0 -1] is consistent.
False
The product of two elementary matrices of the same size must be an elementary matrix.
False
The sum of two invertible matrices of the same size must be invertible.
False
Two nxn matrices, A and B, are inverses of one another if and only if AB = AB = 0.
False
(A±B)^T = A^T ± B^T
True
(kA)^T = kA^T
True
A linear system whose equations are all homogeneous must be consistent.
True
A single linear equation with two or more unknowns must have infinitely many solutions.
True
A square matrix containing a row or column of zeros cannot be invertible.
True
A square matrix with a row or column of zeros is singular.
True
All leading 1's in a matrix in REF must occur in different columns.
True
Any matrix A times the identity matrix equals A.
True
Elementary row operations permit one row of an augmented matrix to be subtracted from another.
True
Every elementary matrix is invertible, and the inverse is also an elementary matrix.
True
Every elementary matrix is invertible.
True
Every matrix has a unique RREF.
True
For every matrix A, it is true that (A^T)^T = A.
True
For every square matrix A, it is true that tr(A^T) = tr(A).
True
If A and B are matrices of the same size and k is a constant, then (kA+B)^T = kA^T + B^T
True
If A and B are row equivalent, and if B and C are row equivalent, then A and C are row equivalent.
True
If A is an (n×n) matrix and c is a scalar, then tr(cA) = c tr(A)
True
If A is an invertible matrix, then A transpose is also invertible and the inverse of the transpose equals the transpose of the inverse.
True
If A is an invertible matrix, then so is A^T.
True
If A is an nxn matrix that is not invertible, then the linear system Ax=0 has infinitely many solutions.
True
If A is an nxn matrix that is not invertible, then the matrix obtained by interchanging two rows of A cannot be invertible.
True
If A is invertible and a multiple of the first row of A is added to the second row, then the resulting matrix is invertible.
True
If A, B, and C are matrices of the same size such that A - C = B - C, then A = B.
True
If AB+BA is defined, then A and B are square matrices of the same size.
True
If a homogeneous linear system of n equations in n unknowns has a corresponding augmented matrix with a RREF containing n leading 1's, then the linear system has only the trivial solution.
True
If a matrix is in RREF, then it is also in REF.
True
If a product of matrices is singular, then at least one of the factors must be singular.
True
The RREF and all REF's of a matrix have the same number of zero rows.
True
The following matrix has no main diagonal: [1 2 3 4 5 6]
True
The linear system x - y = 3 2x - 2y = k cannot have a unique solution, regardless of the value of k.
True
The transpose of a product of any number of matrices is the product of the transposes in the reverse order.
True
