Question

Let P,Q be square matrices of order n, and let u1​,u2​,…,un​ be column vectors in Rⁿ. Which of the following statements are always true? (There may be multiple answers.)

A. If P,Q are orthogonal, then P+Q is also orthogonal.

B. If P,Q are orthogonal, then PQ is also orthogonal.

C. If {u1​,u2​,…,un​} is orthogonal, then (u1​u2​…un​) is an orthogonal matrix.

D. If {u1​,u2​,…,un​} is orthonormal, then (u1​u2​…un​) is an orthogonal matrix.

Answer 1

Statement B and D are always true in regard to orthogonal matrices and vectors.

Step-by-step explanation:

Statement B is always true. If P and Q are orthogonal matrices, then their product PQ is also orthogonal. This is because the product of orthogonal matrices preserves the orthogonality property.

Statement D is always true. If the column vectors {u1, u2, ..., un} are orthonormal, meaning they are both orthogonal and normalized, then the matrix formed by arranging these column vectors as columns, (u1 u2 ... un), is an orthogonal matrix. This is because each column vector is orthogonal to every other column vector and has a magnitude of 1.