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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 .
Which matrices are symmetric? 1 2 −3
A= 2 3
B = 0 −2 0 0
C = −2 −2
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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