Alg1131W12T4 - Transposes

Sunday, 27 May 2018

10:05 PM

The transpose of the 3 x 4 matrix is the 4 x 3 matrix 2 —1 17 2 UT _ 7 6 o 7 -3 2 —1 -3 o 5 6 0 9 17 2 9 4 obtained by interchanging the rows and the columns of U. A square matrix S is called symmetric if ST 5 4 S and skew-symmetric if ST and B then . AT . B+BT Recall: the Maple syntax for the matrix

 

ii) If A and B are square matricies then (AB)T is C) equal to AT BT O undefined. iii) IfC = (1 3) then the matrix CCT C) the same as CCT @ a square matrix O a vector O undefined. Note • the Maple notation for the matrix 112 >,< 314 >> • the Maple syntax for a 1 x 1 matrix ( 1 ) is << 1 cTc is

 

For a general matrix A c is a skew-symmetric ' we can see that A + AT d O matrix. symmetric O matrix; while A AT Now since A = —(A + AT) + J (A — AT), we can deduce that every square matrix A can be written as O the sum of two symmetric matrices @ the sum of a symmetric and a skew-symmetric matrix O the sum of two skew-symmetric matrices. Note: • the Maple notation for the matrix alb >,< old • include a space between the less than symbol < and any letters if you want your answer to display correctly.

 

Give an example of a 2 x 2 matrix which is both symmetric and skew-symmetric: 1 Note: the Maple notation for the matrix 3 2 112 >,< 314 4

 

True or False The product of 2 x 2 symmetric matricies is itself symmetric False The product of a symmetric matrix and a skew-symmetic matrix is skew-symmetric False • A skew-symmetric square matrix always has zeros along its diagonal. True • For any ftvo square matricies A and B, the matrix AB — BTAT is skew-symmetric True • If A and B are symmetric matrices, then so is AB False • If A and B are skew-symmetric matrices, then so is AB True • If A and B are symmetric matrices, then AB True BA is skew symmetric

 

 

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