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aivan3 [116]
3 years ago
5

Is the inverse of a symmetric matrix also symmetric?

Mathematics
1 answer:
Ahat [919]3 years ago
7 0
Yes, the inverse of a symmetric matrix is also symmetric.

Take the symmetric matrix A, we have:

AA^{-1} = I

and

I^{T} = I

This gives:

(AA^{-1})^{T} = AA^{-1}

Using the properties:

AA^{-1} = A^{-1}A and (AA^{-1})^{T} = (A^{-1})^{T}A^{T}

We get:

(A^{-1})^{T}A^{T} = A^{-1}A

Since A^{T} = A, we can perform the substitution to get:

(A^{-1})^{T}A = A^{-1}A

Multiplying by A^{-1} on both sides:

(A^{-1})^{T}AA^{-1} = A^{-1}AA^{-1}

(A^{-1})^{T}I = A^{-1}I

(A^{-1})^{T} = A^{-1}

Proving that the inverse of a symmetric matrix is also symmetric.
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Step-by-step explanation:

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\frac{3}{5}^1^2/\frac{3}{5}^5=\frac{2187}{78125}

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\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{b}\cdot \frac{c}{d}=\frac{a\:\cdot \:c}{b\:\cdot \:d}

\frac{5*2187}{256*78125}

\mathrm{Factor\:the\:number:\:}\:78125=5\cdot \:15625

=\frac{5*2187}{256*5*15625}

\mathrm{Cancel\:the\:common\:factor:}\:5

=\frac{2187}{256*15625}

\mathrm{Multiply\:the\:numbers:}\:256\cdot \:15625=4000000

\frac{2187}{4000000}

[RevyBreeze]

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