MAT 22A true/false questions

A diagonal matrix is invertible if and only if all of its diagonal entries are positive.

False

An expression of an invertible matrix A as a product of ele- mentary matrices is unique.

False

An m × n matrix has m column vectors and n row vectors.

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 A2 −B2 =(A−B)(A+B).

False

For all square matrices A and B of the same size, it is true that (A+B)^2 =A^2 +2AB+B^2.

False

If A and B are 2×2 matrices, then AB=BA.

False

If A and B are invertible matrices of the same size, then AB is invertible and (AB)^−1 = A^−1(B^−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 n×n matrices such that A+B is symmetric, then A and B are symmetric.

False

If A and B are n×n matrices such that A+B is upper triangular, then A and B are upper triangular.

False

If A and B are square matrices of the same order, then (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 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 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 row echelon form has a leading 1, then all entries that are not leading 1's are zero.

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

The i th row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B.

False

The inverse of an invertible lower triangular matrix is an upper triangular matrix.

False

The sum of two invertible matrices of the same size must be invertible.

False

The transpose of an upper triangular matrix is an upper tri- angular matrix.

False

Two n × n matrices, A and B, are inverses of one another if and only if AB = BA = 0.

False

f A^2 is a symmetric matrix, then A is a symmetric matrix.

False

if p(x)=a0 +a1 x+a2 x^2 +···+am x^m 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 number of unknowns, then the system must be inconsistent.

False; a system is inconsistent if It has no solutions

The linear system with corresponding augmented matrix [2 −1 4 ] [0 0 −1] is consistent

False; the metric here is inconsistent bc 0 does not equal -1

Multiplying a row of an augmented matrix through by zero is an acceptable elementary row operation

False; you can only multiply a row with a nonzero constant

A homogeneous linear system in n unknowns whose corre- sponding 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 matrix that is both symmetric and upper triangular must be a diagonal matrix.

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

All entries of a symmetric matrix are determined by the entries occurring on and above the main diagonal.

True

All entries of an upper triangular matrix are determined by the entries occurring on and above the main diagonal.

True

All leading 1's in a matrix in row echelon form must occur in different columns.

True

Elementary row operations permit one row of an augmented matrix to be subtracted from another.

True

Every elementary matrix is invertible.

True

For every matrix A, it is true that (AT )T = A.

True

For every square matrix A, it is true that tr(AT ) = 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 matrices, then the linear systems Ax = 0 and Bx = 0 have the same solution set.

True

If A and B are row equivalent, and if B and C are row equiv- alent, then A and C are row equivalent.

True

If A is a 6×4 matrix and B is an m×n matrix such that BTAT is a 2×6 matrix ,then m=4 and n=2.

True

If A is a square matrix, and if the linear system Ax = b has 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 A^T .

True

If A is an n × n matrix that is not invertible, then the linear system Ax = 0 has infinitely many solutions.

True

If A is an n × n matrix that is not invertible, then the matrix obtained by interchanging two rows of A cannot be invertible.

True

If A is an n×n matrix and c is a scalar, ,then tr(cA) = c tr(A).

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 B has a column of zeros, then so does AB if this product is defined.

True

If a homogeneous linear system of n equations in n unknowns has a corresponding augmented matrix with a reduced row echelon form containing n leading 1's, then the linear system has only the trivial solution.

True

If a matrix is in reduced row echelon form, then it is also in row echelon form.

True

If kA is a symmetric matrix for some k does not equal 0, then A is a symmetric matrix.

True

It is impossible for a system of linear equations to have exactly two solutions.

True

Let A and B be n×n matrices. If A or B(or both) are not invertible, then neither is AB.

True

Let A be an n×n matrix and S is an n×n invertible matrix. If x is a solution to the linear system (S−1AS)x = b, then Sx is a solution to the linear system Ay = Sb.

True

Let A be an n×n matrix. The linear system Ax=4x has a unique solution if and only if A − 4I is an invertible matrix.

True

The linear system x− y=3 2x − 2y = k cannot have a unique solution, regardless of the value of k.

True

The matrix [1 2 3] [4 5 6] has no main diagonal

True

The matrix A= [a b] [c d] is invertible if and only if ad-bc does not equal zero.

True

The sum of a diagonal matrix and a lower triangular matrix is a lower triangular matrix.

True

The transpose of a diagonal matrix is a diagonal matrix.

True

The product of two elementary matrices of the same size must be an elementary matrix.

false

The sum of an upper triangular matrix and a lower triangular matrix is a diagonal matrix.

false

If A and B are n×n matrices such that AB=In, then BA = In.

true