math test 2 true false

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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


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