MATA22 - True or False
A square matrix is nonsingular if and only if its determinant is positive
False
An n x n matrix is diagonalizable if and only if it has n distinct eigenvalues
False
Every complex number has two distinct square roots in C
False
Every function mapping R^n into R^m is a linear transformation
False
Every invertible matrix is diagonalizable
False
Every n x n matrix has n distinct (possibly complex) eigenvalues
False
Every n x n matrix is diagonalizable
False
Every square matrix has real eigenvalues
False
Every triangular matrix is diagonalizable
False
Every vector in a vector space V is an eigenvector of the identity transformation of V into V
False
For any vector a in R^3, we have ||a x a|| = ||a||^2
False
If (a + bi)^3 = 8, then a^2 + b^2 = 4
False
If Arg(z) = 3π/4 and Arg(w) = -π/2, then Arg(z/w) = 5π/4
False
If T and T' are different linear transformations mapping R^n into R^m, then we may have T(ei) = T'(ei) for all standard basis vectors ei of R^n
False
If a matrix A is multiplied by a scalar c, the determinant of the resulting matrix is c * det(A)
False
If an n x n matrix A is diagonalizable, there is a unique diagonal matrix D that is similar to A
False
If det(A) = 2 and det(B) = 3, then det(A + B) = 5
False
If two rows and also two columns of a square matrix A are interchanged, the determinant changes sign
False
If λ is an eigenvalue of a matrix A, then λ is an eigenvalue of A + cI for all scalars c
False
In order for the determinant of a 3 x 3 matrix to be zero, two rows of the matrix must be parallel vectors in R^3
False
The Fundamental Theorem of Algebra asserts that the algebraic operations of addition, subtraction, multiplication, and division are possible with any two complex numbers, as long as we do not divide by zero
False
The determinant det(A) is defined for any matrix A
False
The determinant of a 2 x 2 matrix is a vector
False
The determinant of a 3 x 3 matrix is zero if the points in R^3 given by the rows of the matrix lie in a plane
False
The determinant of a square matrix is the product of the entries on its main diagonal
False
The existence of complex numbers is more doubtful than the existence of real numbers
False
The formula A^-1 = (1/det(A))adj(A) is of practical use in computing the inverse of a large nonsingular matrix
False
The parallelogram in R^2 determined by nonzero vectors a and b is a square if and only if a * b = 0
False
The product of a square matrix and its adjoint is the identity matrix
False
The product of two complex numbers cannot be a real number unless both numbers are themselves real or unless both are of the form bi, where b is a real number
False
The same matrix may be the standard matrix representation for several different linear transformations
False
The square of every complex number is a positive real number
False
There can only be one eigenvector associated with an eigenvalue of a linear transformation
False
A homogeneous square linear system has a nontrivial solution if and only if the determinant of its coefficient matrix is zero
True
A linear transformation having an m x n matrix as standard matrix representation maps R^n into R^m
True
An invertible linear transformation mapping R^n into itself has a unique inverse
True
An n x n matrix is diagonalizable if and only if the algebraic multiplicity of each of its eigenvalues equals the geometric multiplicity
True
Composition of linear transformations corresponds to multiplications of their standard matrix representations
True
Every linear transformation is a function
True
Every n x n matrix has n not necessarily distinct and possibly complex eigenvalues
True
Every n x n real symmetric matrix is real diagonalizable
True
Every nonzero complex number has two distinct square roots in C
True
Every nonzero vector in a vector space V is an eigenvector of the identity transformation of V into V
True
For every square matrix A, we have det(AA^T) = det(A^TA) = [det(A)]^2
True
Function composition is associative
True
If A and B are similar square matrices and A is diagonalizable, then B is also diagonalizable
True
If A and B are similar square matrices, then det(A) = det(B)
True
If B = {b1, b2, ..., bn} is a basis for R^n and T and T' are linear transformations mapping R^n into R^m, then T(x) = T'(x) for all x∈R^n if and only if T(bi) = T'(bi) for i = 1, 2, ..., n
True
If T and T' are different linear transformations mapping R^n into R^m, then we may have T(ei) = T'(ei) for some standard basis vector ei of R^n
True
If an n x n matrix A is multiplied by a scalar c, the determinant of the resulting matrix is c^n * det(A)
True
If an n x n matrix has n distinct real eigenvalues, it is diagonalizable
True
If det(A) = 2 and det(B) = 3, then det(AB) = 6
True
If the angle between vectors a and b in R^3 is π/4, then ||a x b|| = |a * b|
True
If two rows of a 3 x 3 matrix are interchanged, the sign of the determinant is changed
True
If v is an eigenvector of a matrix A, then v is an eigenvector of A + cI for all scalars c
True
If v is an eigenvector of an invertible matrix A, then cv is an eigenvector of A^-1 for all nonzero scalars c
True
If z + z conjugate = 2z, then z is a real number
True
Pencil-and-paper computations with complex numbers are more cumbersome than with real numbers
True
The box in R^3 determined by vectors a, b and c is a cube if and only if a * b = a * c = b * c = 0 and a * a = b * b = c * c
True
The column vectors of an n x n matrix are independent if and only if the determinant of the matrix is nonzero
True
The determinant det(A) is defined for each square matrix A
True
The determinant of a 3 x 3 matrix is zero if the points in R^3 given by the rows of the matrix lie in a plane through the origin
True
The determinant of a 3 x 3 matrix is zero if two rows of the matrix are parallel vectors in R^3
True
The determinant of a lower-triangular square matrix is the product matrix is the product of the entries on its main diagonal
True
The determinant of a square matrix is a scalar
True
The determinant of an elementary matrix is nonzero
True
The determinant of an upper-triangular square matrix is the product of the entries on its main diagonal
True
The product of a square matrix and its adjoint is equal to some scalar times the identity matrix
True
The transpose of the adjoint of A is the matrix of cofactors of A
True
There can be only one eigenvalue associated with an eigenvector of a linear transformation
True