Solution. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.. C. Consider the linear system x + 2y + 3z = a 2x − y + z = b is also symmetric. A symmetric n × n A matrix always has n distinct real eigenvalues. MATH 54 TRUE/FALSE QUESTIONS FOR MIDTERM 2 SOLUTIONS PEYAM RYAN TABRIZIAN 1. Let us look into some problems to understand the concept. Question 1 : Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix: The next leaﬂets in the series will show the conditions under which we can add, subtract and multiply matrices. Any square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. D. If A is symmetric, then A + A2 is symmetric. for all indices and .. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Transcript. Note that whereas C is a 3× 2 matrix, its transpose, CT, is a 2× 3 matrix. (d) The eigenvector matrix Sof a symmetric matrix is symmetric. E. The sum A + AT is always symmetric. Ask Questions, Get Answers Menu X. home ask tuition questions practice papers mobile tutors pricing C. If A is skew symmetric, then A3 is symmetric. Check Answer and Solu (a) TRUE If Ais diagonalizable, then A3 is diagonalizable. Exercise 24.4. Suppose A is an n x n symmetric matrix. COMEDK 2005: If A is a square matrix.such that A3 = 0, then (I + A)-1 is (A) I - A (B) I - A-1 (C) I - A + A2 (D) I + A + A2. (Hint: if you are stuck, look back at Example 20.3.6.) (b) Using the expression A = WDW-1, show that A is invertible exactly when its eigenvalues are all nonzero. More generally, if C is an m× n matrix, its transpose, CT, is a n× m matrix. FALSE( - They need not be distinct) A quadratic form has no cross-product terms if and only if the matrix of the quadratic form is a diagonal matrix. If A is a matrix of order m x n and B is a matrix of order n x p then the order of AB is: [ a b c ] is a Let A be a square matrix. Thus, any symmetric matrix must be diagonalizable.) Misc 5 Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric. (a) alse.F orF example, A= [0 1 0 0]. B. (a) Explain why each of A², A3, etc. (b) alse.F orF example, the matrix A= [0 1 0 0] has one eigenvector, but is not symmetric. then CT = 7 −3 4 1 2 4!. (A= PDP 1, so A3 = PD3P= PeDePe1, where Pe= Pand De= D3, which is diagonal) (b) TRUE If Ais a 3 3 matrix with 3 (linearly independent) If A and B are matrices of same order, then (AB’ – BA’) is a A. skew symmetric matrix B. null matrix C. symmetric matrix D. unit matrix asked Sep 18 in Matrices by Shyam01 ( 50.3k points) matrices Then, we can write. The product AAT is always symmetric. If A is a square matrix, then show that (a) (A + AT) is symmetric matrix. 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