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Problem 16

Can the matrix $E = \begin{array}{ll}\mid 3 & 1 \mid \\ \mid 1 & -1 \mid\end{array}$ be written as a linear combination of the matrices $\begin{array}{rl}\mathrm{m}_{1}=\mid 1 & 1\left|, \quad \mathrm{~m}_{2}=\right| 0 & 0 \mid & \mathrm{m}_{3}=\mid 0 & 2 \\ \mid 1 & 0 \mid & \mid 1 & 1 \mid & \mid 0 & -1 \mid\end{array}$ and \(\begin{array}{rc}\mathrm{m}_{4}=\mid 0 & 1 \mid \\ \mid 1 & 0\end{array}\).

Expert verified

Yes, the matrix \(E = \begin{bmatrix} 3 & 1 \\ 1 & -1 \end{bmatrix}\) can be written as a linear combination of the matrices \(m_1, m_2, m_3,\) and \(m_4\). The linear combination is:
\(E = 3 \cdot m_1 - 1 \cdot m_2 + 0 \cdot m_3 - 1 \cdot m_4 = \begin{bmatrix} 3 & 1 \\ 1 & -1 \end{bmatrix}\)

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