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Transpusa unei matrice

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Dacă avem matricea:

A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} &  \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}, \!

transpusa acesteia este:

{}^tA =  \begin{pmatrix} a_{11} & a_{21} & \cdots & a_{m1} \\ a_{12} & a_{22} &  \cdots & a_{m2} \\ \cdots & \cdots & \cdots & \cdots \\ a_{1n} & a_{2n} & \cdots & a_{mn} \end{pmatrix}. \!

O matrice egală cu transpusa ei se numeşte matrice simetrică.

Exemplu: Fie

A = \begin{pmatrix} 1 &  2 &  3  & 4 \\ 2 &  2  & 3  & 4 \\ 3 & 3 & 3 & 4 \\ 4 & 4 & 4 & 4  \end{pmatrix} \!

Atunci:

{}^t A = \begin{pmatrix} 1 &  2 & 3  & 4 \\ 2 &  2  & 3  & 4 \\ 3 & 3 & 3 & 4 \\ 4 & 4 & 4 & 4  \end{pmatrix} \!
 \Rightarrow \; {}^t A = A \!

Proprietăţi Edit

1. {}^t ({}^t A) = A. \!

2. {}^t (A+B) = {}^t A + {}^t B. \!

3. t(A x B)=tB x tA

4. {}^t (\lambda A) = \lambda \; {}^t A. \!

5. det ({}^t A) = det (A). \!

6. ({}^t A)^{-1} = {}^t (A^{-1}). \!

Resurse Edit

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