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Express the matrix A = \( \begin{bmatrix} 3 & -4 \\[0.3em] 1& -1 \\[0.3em] \end{bmatrix}\) as the sum of a symmetric matrix and a skew-symmetric matrix.

A = [(3,-4)(1,-1)]

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Given A = \( \begin{bmatrix} 3 &-4 \\[0.3em] 1& -1 \\[0.3em] \end{bmatrix}\), to express as the sum of symmetric matrix P and skew symmetric matrix Q.

A = P + Q

Where P = \(\frac{1}{2}\)(A + A') and Q = \(\frac{1}{2}\)(A - A'), we will find transpose of matrix A

A' = \( \begin{bmatrix} 3 &1 \\[0.3em] -4& -1 \\[0.3em] \end{bmatrix}\)

Now using the above formulas

Hence A = P + Q

\(\Rightarrow\) \( \begin{bmatrix} 3 &-4 \\[0.3em] 1& -1 \\[0.3em] \end{bmatrix}\)

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