The formula ker(A) = ker(AT A) holds for all matrices A

If T is a linear transformation from Rn to Rn such that T (e1), T (e2), . . . , T (en) are all unit vectors, then T must be an orthogonal transformation. 2

If A is an invertible matrix, then the equation (AT )1 = (A1)T must hold.

If matrix A is orthogonal, then matrix A2 must be orthogonal as well.

The equation (AB)T = AT BT holds for all n n matrices A and B.

If A and B are symmetric n n matrices, then A + B must be symmetric as well.

If matrices A and S are orthogonal, then S1AS is orthogonal as well.

. All nonzero symmetric matrices are invertible.

If A is an n n matrix such that AAT = In, then A must be an orthogonal matrix.

If u is a unit vector in Rn, and L = span(u), then projL (x) = (x u)x for all vectors x in Rn. 10. I

. If A is a symmetric matrix, then 7A must be symmetric as well.

If x and y are two vectors in Rn, then the equation x + y2 = x2 + y2 must hold. 12.

The equation det(AT ) = det(A) holds for all 2 2 matrices A

If matrix A is orthogonal, then AT must be orthogonal as well.

If A and B are symmetric nn matrices, then AB must be symmetric as well.

If matrices A and B commute, then A must commute with BT as well.

If A is any matrix with ker(A) = {0}, then the matrix AAT represents the orthogonal projection onto the image of A.

If A and B are symmetric n n matrices, then ABBA must be symmetric as well.

If matrices A and B commute, then matrices AT and BT must commute as well.

There exists a subspace V of R5 such that dim(V) = dim(V ), where V denotes the orthogonal complement of V.

Every invertible matrix A can be expressed as the product of an orthogonal matrix and an upper triangular matrix.

The determinant of all orthogonal 2 2 matrices is 1.

If A is any square matrix, then matrix 1 2 (A AT ) is skew-symmetric.

The entries of an orthogonal matrix are all less than or equal to 1.

Every nonzero subspace of Rn has an orthonormal basis.

3 4 4 3 is an orthogonal matrix.

If V is a subspace of Rn and x is a vector in Rn, then vector projV x must be orthogonal to vector x projV x. 27

If A and B are orthogonal 2 2 matrices, then AB = B A.

If A is a symmetric matrix, vector v is in the image of A, and w is in the kernel of A, then the equation v w = 0 must hold. 2

The formula ker(A) = ker(AT A) holds for all matrices A

If AT A = AAT for an n n matrix A, then A must be orthogonal.

There exist orthogonal 22 matrices A and B such that A + B is orthogonal as well.

If Axx for all x in Rn, then A must represent the orthogonal projection onto a subspace V of Rn.

If A is an invertible matrix such that A1 = A, then A must be orthogonal.

If the entries of two vectors v and w in Rn are all positive, then v and w must enclose an acute angle. 35

The formula (ker B) = im(BT ) holds for all matrices B.

The matrix AT A is symmetric for all matrices A.

If matrix A is similar to B and A is orthogonal, then B must be orthogonal as well.

The formula im(B) = im(BT B) holds for all square matrices B.

If matrix A is symmetric and matrix S is orthogonal, then matrix S1AS must be symmetric.

If A is a square matrix such that AT A = AAT , then ker(A) = ker(AT ).

Any square matrix can be written as the sum of a symmetric and a skew-symmetric matrix.

. If x1, x2,..., xn are any real numbers, then the inequality 9 n k=1 xk :2 n n k=1 (x2 k ) must hold

If AAT = A2 for a 2 2 matrix A, then A must be symmetric.

If V is a subspace of Rn and x is a vector in Rn, then the inequality x (projV x) 0 must hold. 4

If A is an n n matrix such that Au = 1 for all unit vectors u, then A must be an orthogonal matrix.

If A is any symmetric 2 2 matrix, then there must exist a real number x such that matrix A x I2 fails to be invertible.

There exists a basis of R22 that consists of orthogonal matrices.

f A =  1 2 2 1 , then the matrix Q in the Q R factorization of A is a rotation matrix.

There exists a linear transformation L from R33 to R22 whose kernel is the space of all skew-symmetric 3 3 matrices.

If a 3 3 matrix A represents the orthogonal projection onto a plane V in R3, then there must exist an orthogonal 3 3 matrix S such that ST AS is diagonal.