Linear Algebra Chapter 1 True-False
1.1b) Multiplying a linear equation through by zero is an acceptable elementary row operation.
False
1.1e) If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent.
False
1.1f) 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
1.1h) The linear system with corresponding augmented matrix [ 2 -1 4 ] [ 0 0 -1 ] is consistent.
False
1.2b) 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
1.2c) Every matrix has a unique row echelon form.
False
1.2f) 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
1.2h) 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
1.2i) If a linear system has more unknowns than equations, then it must have infinitely many solutions.
False
1.3L) If A, B, and C are square matrices of the same order such that AC = BC, then A = B
False
1.3b) An m x n matrix has m column vectors and n row vectors.
False
1.3c) If A and B are 2 x 2 matrices then AB = BA
False
1.3d) The i th row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B.
False
1.3f) If A and B are square matrices of the same order, then tr(AB) = tr(A)tr(B).
False
1.3g) If A and B are square matrices of the same order, then (AB)^T = (A^T)(B^T)
False
1.3o) If B has a column of zeros, then so does BA if this product is defined.
False
1.4a) Two n x n matrices, A and B, are inverses of one another if and only if AB = BA = 0.
False
1.4b) For all square matrices A and B of the same size, it is true that (A + B)^2 = A^2 + 2AB + B^2
False
1.4c) For all square matrices A and B of the same size, it is true that A^2 - B^2 = (A - B)(A + B).
False
1.4d) If A and B are invertible matrices of the same size, then AB is invertible and (AB)^-1 = (A^-1)(B^-1)
False
1.4e) If A and B are matrices such that AB is defined, then it is true that (AB)^T = (A^T)(B^T)
False
1.4i) 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
1.4k) The sum of two invertible matrices of the same size must be invertible.
False
1.5a) The product of two elementary matrices of the same size must be an elementary matrix.
False
1.5g) An expression of the invertible matrix A as a product of elementary matrices is unique.
False
1.7L) If A^2 is a symmetric matrix, then A is a symmetric matrix
False
1.7b) The transpose of an upper triangular matrix is an upper triangular matrix.
False
1.7c) The sum of an upper triangular matrix and a lower triangular matrix is a diagonal matrix.
False
1.7f) The inverse of an invertible lower triangular matrix is an upper triangular matrix.
False
1.7g) A diagonal matrix is invertible if and only if all of its diagonal entries are positive.
False
1.7j) If A and B are n x n matrices such that A + B is symmetric, then A and B are symmetric.
False
1.7k) If A and B are n x n matrices such that A + B is upper triangular, then A and B are upper triangular.
False
1.1a) A linear system whose equations are all homogeneous must be consistent.
True
1.1c) The linear system x - y = 3 2x - 2y = k cannot have a unique solution, regardless of the value of k.
True
1.1d) A single linear equation with two or more unknowns must always have infinitely many solutions
True
1.1g) Elementary row operations permit one equation in a linear system to be subtracted from another.
True
1.2a) If a matrix is in reduced row echelon form, then it is also in row echelon form.
True
1.2d) A homogeneous 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
1.2e) All leading 1's in a matrix in row echelon form must occur in different columns.
True
1.2g) 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
1.3a) The matrix [ 1 2 3 ] [4 5 6 ] has no main diagonal
True
1.3e) For every matrix A, it is true that (A^T)^T = A
True
1.3h) For every square matrix A, it is true that tr(A^T) = tr(A)
True
1.3i) f A is a 6 x 4 matrix and B is an m x n matrix such that (B^T)(A^T) is a 2 x 6 matrix, then m=4 and n=2.
True
1.3j) If A is an n x n matrix and c is scalar, then tr(cA) and c(tr(A)).
True
1.3k) If A, B, and C are matrices of the same size such that A - C = B - C, then A = B .
True
1.3m) If AB + BA is defined, then A and B are square matrices of the same size.
True
1.3n) If B has a column of zeros, then so does AB if this product is defined.
True
1.4f) The matrix A = [ a b ] [ c d ] is invertible if and only if ad - bc does not equal 0
True
1.4h) If A is an invertible matrix, then so is A^T.
True
1.4j) A square matrix containing a row or column of zeros cannot be invertible.
True
1.5b) Every elementary matrix is invertible.
True
1.5c) If A and B are row equivalent, and if B and C are row equivalent, then A and C are row equivalent.
True
1.5d) If A is an n x n matrix that is not invertible, then the linear system Ax = 0 has infinitely many solutions.
True
1.5e) If A is an n x n matrix that is not invertible, then the matrix obtained by interchanging two rows of A cannot be invertible.
True
1.5f) 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
1.6a) It is impossible for a linear system of linear equations to have exactly two solutions.
True
1.6b) If the linear system Ax = b has a unique solution, then the linear system Ax = c also must have a unique solution.
True
1.6c) If A and B are n x n matrices such that AB = Identity matrix n, then BA = Identity matrix n
True
1.6d) If A and B are row equivalent matrices, then the linear systems Ax = 0 and Bx = 0 have the same solution set.
True
1.6e) If A is an n x n matrix and S is an n x n invertible matrix, then if x is a solution to the linear system (S^-1(AS))x = b, then Sx is a solution to the linear system Ay = Sb
True
1.6f) Let A be an n x n matrix. The linear system Ax = 4x has a unique solution if an only if A - 4(subscript Identity) is an invertible matrix
True
1.6g) Let A and B be n x n matrices. If A or B (or both) are not invertible, then neither is AB.
True
1.7a) The transpose of a diagonal matrix is a diagonal matrix.
True
1.7d) All entries of a symmetric matrix are determined by the entries occurring on and above the main diagonal.
True
1.7e) All entries of an upper triangular matrix are determined by the entries occurring on and above the main diagonal.
True
1.7h) The sum of a diagonal matrix and a lower triangular matrix is a lower triangular matrix.
True
1.7i) A matrix that is both symmetric and upper triangular must be a diagonal matrix.
True
1.7m) If kA is a symmetric matrix for some k not equal to 0, then A is a symmetric matrix.
True
14g) If A and B are matrices of the same size and k is a constant, then (kA + B)^T = kA^T + B^T
True
