math test 2 true false
A diagonal matrix is invertible if and only if all of its diagonal entries are positive.
False
A square matrix A is invertible if and only if det(A) = 0.
False
Collinear vectors with the same length are equal.
False
Every vector in Rn has a positive norm.
False
For all square matrices A and B, it is true that det(A + B) = det(A) + det(B)
False
For every square matrix A and every scalar c, it is true that det(cA) = c det(A).
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 size such that det(A) = det(B), then det(A + B) = 2 det(A).
False
If A has a row of zeros, then so does adj(A).
False
If A is a 3 × 3 matrix and B is obtained from A by adding 5 times the first row to each of the second and third rows, then det(B) = 25 det(A).
False
If A is a 3 × 3 matrix, then det(2A) = 2 det(A).
False
If A is an n × n matrix and B is obtained from A by multiplying each row of A by its row number, then det(B) = ((n(n + 1))/2) det(A)
False
If A2 is a symmetric matrix, then A is a symmetric matrix.
False
If a and b are scalars such that au + bv = 0, then u and v are parallel vectors.
False
If k and m are scalars and u and v are vectors, then (k + m)(u + v) = ku + mv
False
If k is a scalar and v is a vector, then v and kv are parallel if and only if k ≥ 0.
False
If u · v = 0, then either u = 0 or v = 0.
False
If u · v = u · w, then v = w.
False
The determinant of a lower triangular matrix is the sum of the entries along the main diagonal.
False
The determinant of the 2 × 2 matrix a b c d is ad + bc.
False
The expressions (u · v) + w and u · (v + w) are both meaningful and equal to each other.
False
The inverse of an invertible lower triangular matrix is an upper triangular matrix.
False
The linear combinations a1v1 + a2v2 and b1v1 + b2v2 can only be equal if a1 = b1 and a2 = b2.
False
The sum of an upper triangular matrix and a lower triangular matrix is a diagonal matrix.
False
The transpose of an upper triangular matrix is an upper triangular matrix.
False
The vectors (a, b) and (a, b, 0) are equivalent.
False
Two equivalent vectors must have the same initial point.
False
Two square matrices that have the same determinant must have the same size.
False
A matrix that is both symmetric and upper triangular must be a diagonal matrix.
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
For all vectors u, v, and w in Rn, we have ||u+v+w|| <= ||u|| + ||v|| + ||w||
True
For every 2 × 2 matrix A it is true that det(A^2) = (det(A))^2.
True
For every n × n matrix A, we have A · adj(A) = (det(A))In
True
If (a, b, c) + (x, y, z) = (x, y, z), then (a, b, c) must be the zero vector.
True
If A and B are n × n matrices such that AB = In, then BA = In.
True
If A and B are square matrices of the same size and A is invertible, then det(A^−1BA) = det(B)
True
If A is a 3 × 3 matrix and B is obtained from A by multiplying the first column by 4 and multiplying the third column by 3/4 , then det(B) = 3 det(A).
True
If A is a 3 × 3 symmetric matrix, thenCij = Cji for all i and j .
True
If A is a 4 × 4 matrix and B is obtained from A by interchanging the first two rows and then interchanging the last two rows, then det(B) = det(A).
True
If A is a square matrix and the linear system Ax = 0 has multiple solutions for x, then det(A) = 0.
True
If A is a square matrix whose minors are all zero, then det(A) = 0.
True
If A is a square matrix with two identical columns, then det(A) = 0.
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 the linear system Ax = 0 has only the trivial solution if and only if the linear system A^−1x = 0 has only the trivial solution.
True
If A is an n × n matrix and there exists an n × 1 matrix b such that the linear system Ax = b has no solutions, then the reduced row echelon form of A cannot be In.
True
If A is invertible, then adj(A) must also be invertible.
True
If E is an elementary matrix, then Ex = 0 has only the trivial solution.
True
If each component of a vector in R3 is doubled, the norm of that vector is doubled.
True
If kA is a symmetric matrix for some k = 0, then A is a symmetric matrix.
True
If the sum of the second and fourth row vectors of a 6 × 6 matrix A is equal to the last row vector, then det(A) = 0.
True
If the vectors v and w are given, then the vector equation 3(2v − x) = 5x − 4w + v can be solved for x.
True
If u + v = u + w, then v = w.
True
If v is a nonzero vector inRn, there are exactly two unit vectors that are parallel to v.
True
If ||u|| = 2, ||v|| = 1, and u · v = 1, then the angle between u and v is π/3 radians.
True
IfAandB are row equivalent matrices, then the linear systems Ax = 0 and Bx = 0 have the same solution set.
True
In R2, if u lies in the first quadrant and v lies in the third quadrant, then u · v cannot be positive.
True
In R2, the vectors of norm5 whose initial points are at the origin have terminal points lying on a circle of radius 5 centered at the origin.
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 matrix of cofactors of A is precisely [adj(A)]T .
True
The minor Mij is the same as the cofactor Cij if i + j is even.
True
The number obtained by a cofactor expansion of a matrix A is independent of the row or column chosen for the expansion.
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 vectors v + (u + w) and (w + v) + u are the same.
True
