3 1. 132 points] For each of the following statements, indicate if it is true or

Question

3 1. 132 points] For each of the following statements, indicate if it is true or false. A correct answer will get 2 points, but a wrong answer will get -2 points. No answer will get 0 point. (a) If a system of linear equations has no free variable, then it has a unique solution. (b) Left-multiplying a matrix A by a diagonal matrix D with nonzero entries on the diagonal, i.e. DA, scales the rows A. (c) det (AAT) 2 0 for any matrixA (d) A square, non-invertible matrix A always have an eigenvalue of 0. (e) An n × n diagonalizable matrix has n distinct eigenvalues. (f) If two vectors u, v E Rmx1 are eigenvectors of a matrix A e Rnxn, then u + v is also arn eigenvector of A. (8) A square matrix A is diagonalizable if and only if the algebraic multiplicity is the same as the geometric multiplicity for each distinct eigenvalue. (h) Each eigenvector of an invertible matrix A is also an eigenvector of A-1 (i) An invertible matrix is diagonalizable. G) The normal equation may have no solution at all. at B (k) If A and B are 3 x 3 and Aa then ABa B J (1) det A2 2 det A. (m) Similar matrices shares the same eigenvectors but not the eigenvalues. (n) An eigenvector cannot be a zero vector. (o) The null space of A is the same as that of AT A (p) An eigenvectors of an n × n diagonalizable matrix forms a basis of Rn.

Explanation / Answer

FALSE

FALSE

TRUE

TRUE

FALSE

TRUE

FALSE

TRUE

TRUE

TRUE

TRUE

TRUE

FALSE

FALSE

TRUE

TRUE

TRUE

Related Questions

Navigate

• Browse (All)
• Browse 3
• Subjects

---

---

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