Math 54 Midterm 2 - Weekend Quizzes

For which of the following matrices A does the sequence of matrices An converge to the zero matrix as n goes to infinity? A. [ 0.8 0.3 0.5 0.6] B. [ 0.8 0.3 -0.5 0.6] C. [ 0.8 0.3 0.6 0.5] D. All of the above. E. None of the above.

B

If A is an mxn matrix and B is an nxm matrix, we have Det(AB) = Det(BA) provided the following condition holds. (Select the answer that avoids unnecessary restrictions. This is a bit tricky and you may need to try out some small examples.) A. Always true, no conditions needed. B. m=n. C. A, B are both square and invertible D. A and B are in upper echelon form.

B

If u is orthogonal to v and v is orthogonal to w, then we must have A. u is orthogonal to w B. ||u + v||² = ||u||² + ||v||² and ||v + w||² = ||v||² + ||w||² C. ||u + v||² = ||u||² + ||v||² and ||u + w||² = ||u||² + ||w||² D. ||u + v||² = ||u||² + ||v||² and ||u + w||² = ||u||² + ||w||² and ||v + w||² = ||v||² + ||w||²

B

In a new basis of R² the new coordinates of the vector [1,3]T are [2,1]T and those of the vector [3,5]T are [4,1]T. What are the coordinates of [1,-1]T in the new basis? A. [1,1]T B. [0,-1]T C. [-1,1]T D.Cannot be determined from the information supplied

B

The determinant of the following matrix is equal to [ 0 0 0 1 0 0 2 1 0 3 2 1 4 3 2 1] A. 1 B. 24 C. -24 D. 0 E. It depends on how we swap rows in the row-reduction

B

We are told that a linear transformation L: R² --> R² has two eigenvalues, 1 and 2, and that the vectors [1,1]T and [1,2]T are eigenvectors for the two eigenvalues (in that order). What is L([0,1]T)? A. [1,2]T B. [1,3]T C. [2,3]T D. [2,4]T

B

Which of the following collections of vectors are orthonormal? A. [2,1,2]^T, [-1, -2, 2]^T, [-2, 2, 1]^T B. [²/₃, ¹/₃, ²/₃]^T, [-¹/₃, -²/₃, ²/₃]^T C. [¹/₃, ²/₃, -²/₃]^T, [-²/₃, ²/₃, ¹/₃]^T, [²/₃, -¹/₃, ²/₃]^T D. [ 1, 1]^T, [-1, 1]^T

B

A 3x3 real matrix A is known to have a complex, non-real eigenvalue. Then, we can conclude that A. A is invertible B. A is a rotation matrix C. A is diagonalizable over the complex numbers D. None of the above

C

Let A be a matrix with linearly independent columns. The least-squares solution s to a (possibly inconsistent) system Ax = b is characterized by the following property: A. s is the shortest vector in the span of b. B. The distance ||s - b|| is as small as possible, and s is in Col(A). C. The distance ||As - b|| is as small as possible. D. b is the orthogonal projection of s onto Col(A)

C

We are told that the square matrix A satisfies the intriguing identity A² = A. We can safely conclude that A. A is invertible B. A is orthogonal C. The only possible eigenvalues of A are 0 and 1 D. A is an orthogonal projection

C

When can we be sure that a real nxn matrix is diagonalizable (over the real numbers)? A.When all of its eigenvalues are real B.When all of its eigenvalues are real and there exists at least one eigenvector for each eigenvalue C.When there are n linearly independent real eigenvectors D.When there are n pivots

C

Which of the following matrices is guaranteed to have non-zero determinant? (Choose all that apply) A. A square matrix whose diagonal entries are all different from zero B. The nxn coefficient matrix of a consistent linear system C. A lower-triangular square matrix whose diagonal entries are all different from zero D. The nxn coefficient matrix of a linear system which has a unique solution E. A square matrix whose entries are all positive

C, D

Which of the following matrices is NOT diagonalizable? A. [ 0 1 -1 5] B. [ 1 1 -1 5] C. [ 2 1 -1 5] D. [ 3 1 -1 5]

D

We are told that the square matrix A is diagonalizable. Pick the matrices below which are NOT guaranteed to be diagonalizable, if any, or else choose option E: A. A^T B. A⁻¹, assuming that A is invertible C. A² D. A + I E. All matrices above are guaranteed to be diagonalizable

E

Cramer's rule is an efficient way to solve a large system of linear equations whose coefficient matrix is invertible.

False

For two nxn matrices A,B, Det(A+B) = Det(A) + Det(B)

False

If A is similar to B and B is an orthogonal matrix, then so is A.

False

If the columns of a 2x2 matrix are orthogonal, then the matrix represents a rotation in R²

False

Similar matrices always have exactly the same eigenvectors.

False

The columns of the nxn matrix S form a basis of Rⁿ if and only if Det(S) = 0

False

The matrix representing a reflection about a line L in R² are diagonalizable only when the line L is the x-axis or the y-axis

False

A 2x2 real matrix with complex (non-real) eigenvalues is diagonalizable over the complex numbers.

True

A square matrix is invertible if and only if 0 is not an eigenvalue

True

If V is a linear subspace of Rⁿ, then the operation which sends vectors in Rⁿ to their orthogonal projection on V is a linear transformation from Rⁿ to itself.

True

If a square matrix has orthonormal columns, then it also has orthonormal rows

True

If the columns of the mxn matrix A are non-zero and pairwise orthogonal, then the system Ax = b has at most one solution.

True

If u, v and w are vectors in R³ and they are all orthogonal to the vector [1,2,3]T, then they are linearly dependent.

True

If ||u - v|| = ||u + v||, then the vectors u and v are orthogonal.

True

Similar matrices have the same characteristic polynomial

True

The determinant of a square matrix is the product of its eigenvalues (included with their multiplicities)

True

The determinant of a square, invertible matrix equals plus or minus the product of its pivots

True

The product of two orthogonal matrices is an orthogonal matrix.

True