6. True or false (10 points, no need to explain your reasoning) 1) Any elementar

Question

6. True or false (10 points, no need to explain your reasoning) 1) Any elementary matrix has a QR decomposition. 2) Any elementary matrix has a LU decomposition. 3) If P2 = P then P is an orthogonal projection matrix. 4) If the null space and column space of a square matrix are orthogonal complements of each other, the matrix must be an orthogonal projection matrix (5) The characteristic polynomial of a 4 x 4 symmetric matrix must have 4 real roots, counting multiplicity (6) If the characteristic polynomial of A is A2(A1 and A is diagonalizable, then the rank of A is 3. (7) The dimension of the orthogonal complement of the column space of a square matrix is the dimeusion of its null space. (8) An orthogonal projection matrix must be orthogonal. 9) The reduced row echelon form of an orthogonal matrix is the identity matrx. (10) Let A and B be square matrices of the same size, AB BA, then if A is symmetric so is B.

Explanation / Answer

(1) true

(2) false ( if the matrix is singular LU decomposition is not accurate)

(3) true

(4) true

(5) false

(6) false

(7) true

(8) false

(9) tue

(10) false

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