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Unformatted text preview: atrix. A is symmetric if and only if
AT = A. In particular, aij = aji for all i , j .
Example
Which matrices are symmetric? 1 2 −3
4 ,
A= 2 3
−3 4
5 1
00
B = 0 −2 0 0
03 1
2 −3
0
C = −2 −2
3
0
3 Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Properties of Symmetric Matrices Theorem
Suppose A = [aij ] and B = [bij ] are symmetric n × n matrices, and
λ is a scalar.
(a) AT is symmetric.
(b) A + B is symmetric.
(c) λA is symmetric.
(d) AB is symmetric if and only if A and B commute.
(e) If A is invertible, then A−1 is symmetric. Outline Diagonal Matrices Triangular Matrices Symmetric Matrices The Proof Proof.
(a) (AT )T = A = AT .
(b) (A + B )T = AT + B T = A + B .
(c) (λA)T = λAT = λA.
(d) (AB )T = B T AT = BA, so AB is symmetric if and only if A
and B commute.
(e) If A is invertible, then so is AT and (A−1...
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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.
 Spring '14
 KenyonJ.Platt
 Linear Algebra, Algebra, Matrices

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