5. Consider the following statements: (i) If Q is an n x n matrix with orthogona

Question

5. Consider the following statements: (i) If Q is an n x n matrix with orthogonal columns with respect to the standard inner product (dot product) on R", then Q is orthogonal matrix (ii) If P and Q are orthogonal matrices then PQ is an orthogonal matrix. (ii) Let A be a 2x2 symmetric matrix with eigenvalues 1 and 2. Let u and v be corresponding eigenvectors respectively. Then the set {u, v} is orthonormal. (iv) If T: R3R3 is a counterclockwise rotation about the positive z-axis through an angle ?, then the standard matrix [T] of T is an orthogonal matrix (v) If A is an n x n symmetric matrix, then PT AP is also symmetric for any nx n How many of the above statements are always TRUE (A) 1 matrix P (B) 2; (C) 3 (D) 4 (E) All of them

Explanation / Answer

4of the above statements are correct so the correct option is D.

A matrix n xn is orthogonal if columns of matrix form orthonormal basis for R^n.so option 1 is correct.

If two matrices are orthogonal then product also orthogonal. So it is correct

A 2 x2 symmetric matrix eigen vectors are orthogonal. So the statement is wrong.

Symmetric matrix means A^T=A

So ( P^T A P)^T= P^T^T A^T P^T= P AP^T

The matrix is symmetric

Related Questions

Navigate

• Browse (All)
• Browse 5
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.