Linear Algebra Exam 2 T or F practice

If A and B are n x n matrices, then (AB)^T = A^T B^T

FALSE — The order reverses: (AB)^T = B^T A^T.

If A and B are invertible n x n matrices, then (AB)^-1 = A^-1 B^-1

FALSE — The order reverses: (AB)^-1 = B^-1 A^-1.

If AB = AC and A is not 0, then B = C

FALSE — Matrix multiplication does not satisfy the cancellation law unless A is invertible.

If A is an n x n matrix and Ax = 0 has a non-trivial solution, then A is non-invertible

TRUE — By IMT, Ax = 0 having only the trivial solution is equivalent to A being invertible.

If A can be row reduced to the identity matrix I_n, then A is invertible

TRUE — By IMT, A ~ I_n if and only if A is invertible.

If A is invertible, then A^T is invertible

TRUE — (A^T)^-1 = (A^-1)^T.

If A is a 2 x 2 matrix with det A = 0, then one column is a scalar multiple of the other

TRUE — det A = 0 means columns are linearly dependent.

Every elementary matrix is invertible

TRUE — Row operations are reversible, so elementary matrices are always invertible.

If the columns of an n x n matrix A span R^n, then they are linearly independent

TRUE — By IMT, for square matrices, spanning R^n and linear independence are equivalent.

If an n x n matrix A is invertible, then Ax = b has a unique solution for each b in R^n

TRUE — x = A^-1 b is the unique solution.

If A is an n x n matrix and Nul A = {0}, then Rank A = n

TRUE — Rank Theorem: Rank A + dim(Nul A) = n.

If T: R^n -> R^n is a one-to-one linear transformation, then T is also onto

TRUE — For an n x n standard matrix, one-to-one and onto are equivalent.

The set V = { [x, y]^T : x >= 0, y >= 0 } is a subspace of R^2

FALSE — Not closed under scalar multiplication (e.g., -1 times a positive vector).

The set H = { [a, b]^T : a + 2b = 0 } is a subspace of R^2

TRUE — Represents a line through the origin in R^2.

The pivot columns of A form a basis for Col A

TRUE — Uses original corresponding columns from matrix A.

The pivot columns of the RREF of A form a basis for Col A

FALSE — You must use the original columns of A, not the reduced matrix columns.

If A is a 3 x 5 matrix, then Nul A is a subspace of R^3

FALSE — Nul A is a subspace of R^5 (number of columns n = 5).

If A is a 4 x 7 matrix, then Rank A + dim(Nul A) = 7

TRUE — Rank Theorem: Rank A + dim(Nul A) = n = 7.

If dim V = p, then any set of p linearly independent vectors in V forms a basis for V

TRUE — Basis Theorem: dim(V) = p means any p independent vectors span V.

For any n x n matrix A, det(-A) = -det(A)

FALSE — det(-A) = (-1)^n det(A). If n is even, det(-A) = det(A).

If A and B are n x n matrices, then det(A + B) = det A + det B

FALSE — Determinants are multiplicative, not additive.

If det A = 3 for a 3 x 3 matrix A, then det(2A) = 24

TRUE — det(2A) = 2^3 * det A = 8 * 3 = 24.

Swapping two rows of a matrix A does not change its determinant

FALSE — Swapping two rows multiplies the determinant by -1.

Adding a scalar multiple of one row to another row of A changes det A

FALSE — Row replacements leave the determinant unchanged.

If det A = 0, then A is non-invertible

TRUE — A matrix is invertible if and only if det A is non-zero.

If A is a triangular matrix, then det A is the sum of the main diagonal entries

FALSE — det A is the product of the main diagonal entries.

det(A^T A) >= 0 for any square matrix A

TRUE — det(A^T A) = det(A^T) * det(A) = (det A)^2 >= 0.