linear algebra true/false
a single linear equation with two or more unknown must have infinitely many solutions
false
an expression of an invertible matrix a as a product of elementary matrices is unique
false
an m x n matrix has m column vectors and n row vectors
false
elementary row operations permit one row of an augmented matrix to be subtracted from another
false
every matrix has a unique row echelon form
false
for all square matrices A and B of the same size, if it 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
for every square matrix Am it is true that tr(At)=tr(A)
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 and B are matrices such that AB is defined, then it is true that (ab)t=atbt
false
if A and B are square matrices of the same order, then tr(AB)=tr(A)tr(B)
false
if A, B, and C are square matrices of the same order such that AC=BC, then A=B
false
if B has a column of zeros, then so does BA if this product is defined
false
if a and b are 2 x 2 matrices, then AB=BA
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 row echelon form, the resulting matrix will still be in row echelon form
false
if every column of a matrix in row echelon form has a leading 1, then all entries that are not leading 1's are zeros
false
if p(x)=ao+a1x+a2x+. . . +amxm and I is an identity matrix, then p(I)=a0+a1+a2+...+am
false
if the number of equations in a linear system exceeds the nmber of unknowns then the system must be consistent
false
if the reduced row echelon form of the augmented matrix for a linear system has a row of zeros, then the system must have infinitely many solutions
false
multiplying a row of an augmented matrix through by zero is an acceptable elementary row operations
false
the ith row vector of a matrix product AB can be computed by multiplying a by the ith row vector of B
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=BA=0
false
If A is an nxn matrix that is not invertible, then the matrix obtained by interchanging two rows of A cannot be invertible
true
Let A and B be nxn matrices. if A or B (or both) are not invertible, then neither is AB
true
a homogenous linear system in n unknowns whose corresponding augmented matrix has a reduced row echelon form with r leading 1's has n-r free variables
true
a linear system whose equations are all homogeneous must be consistent
true
a square matrix containing a row or column or zeros cannot be invertible
true
all leading 1's in a matrix in row echelon form must occur in different columns
true
every elementary matrix is invertible
true
for every matrix, A, it is true that (At)t=A
true
if A and B are matrices of the same size and k is a constant, then (kA+B)t=kAt+Bt
true
if A and B are nxn matrices such that AB=In, then BA=In
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 and B are rpw equivalent matrices, then the linear systems Ax=0 and Bx=0 have the same solution set
true
if A and B are square matrices of the same order, then (AB)t=AtBt
true
if A is a 6x4 matrix an B is an mxn matrix such that BtAt is a 2x6 matrix, then m=4 and n=2
true
if A is a square matrix, and if the linear system Ax=b as a unique solution, then the linear system Ax=c also must have a unique solution
true
if A is an invertible matrix, then so is At
true
if A is an nxn matrix and c is a scalar, then tr(cA)=ctr(a)
true
if A is an nxn matrix that is not invertivle, then the linear system Ax=0 has infinitely many solutions
true
if A, B, and C are matrices of the same size such that A-C=B-C, the A=B
true
if AB=BA is defined, then A and B are square matrices of the same size
true
if B has a column of zeros, then so does AB if this product is defined
true
if a homogenous linear system of n equations in n unknowns has a corresponding augmented marix with a reduced row echelon form containing n leading 1's, then the linear system has only the trivial solution
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 matrix is in reduced row echelon form, then it is also in row echelon form
true
if each equation in a consistent linear system is multiplied through by a consistent c, then all solutions to the new system can be obtained by multiplying solutions from the original system by c
true
it is impossible for a system of linear equations to have exactly two solutions
true
let A be an nxn matrix and s is an nxn invertible matrix. if x is a solution to the linear system (S-1AS)x=b, then Sx is a solution to the linear solution Ay=Sb
true
let A be an nxn matrix, the linear system Ax=4x has a unique solution if an only if A-4i is an invertible matrix
true
the linear system x-y=3 2x-2y=k cannot have a unique solutions, regardless of the value of k
true
the linear system with corresponding augmented matrix 2 -1 4 0 0 -1 is consistent
true
the matrix A=a b c d is invertible if and only if ab-bc is not equal to zero
true
the matrix 1 2 3 4 5 6 has no main diagonal
true