Collection of Maths Problems

Powers and roots of a matrix

Task number: 2664

• Variant 1

  \( \begin{pmatrix} -11 & 30 \\ -10 & 24 \\ \end{pmatrix} \)

• Hint

  The third power of \(\mathbf A=\mathbf R\mathbf D\mathbf R^{-1}\) is straightforwardly \(\mathbf A^3=\mathbf R\mathbf D^3\mathbf R^{-1}\).

  Analogously, the square root is \(\mathbf R\mathbf D^{\frac12}\mathbf R^{-1}\), where \(\mathbf D^{\frac12}\) has on the diagonal square roots of the eigenvalues.

  (In total \(2^n\) solutions on matrices of order \(n\).)

• Resolution

  \( \begin{pmatrix} -11 & 30 \\ -10 & 24 \\ \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ 2 & 1 \\ \end{pmatrix} \begin{pmatrix} 9 & 0 \\ 0 & 4 \\ \end{pmatrix} \begin{pmatrix} -1 & 2 \\ 2 & -3 \\ \end{pmatrix} \)

  \(\begin{pmatrix} 3 & 2 \\ 2 & 1 \\ \end{pmatrix} \begin{pmatrix} 729 & 0 \\ 0 & 64 \\ \end{pmatrix} \begin{pmatrix} -1 & 2 \\ 2 & -3 \\ \end{pmatrix} = \begin{pmatrix} -1931 & 3990 \\ -1330 & 2724 \\ \end{pmatrix} \)

  \(\begin{pmatrix} 3 & 2 \\ 2 & 1 \\ \end{pmatrix} \begin{pmatrix} 3 & 0 \\ 0 & 2 \\ \end{pmatrix} \begin{pmatrix} -1 & 2 \\ 2 & -3 \\ \end{pmatrix} = \begin{pmatrix} -1 & 6 \\ -2 & 6 \\ \end{pmatrix} \)

• Variant 2

  \( \begin{pmatrix} 0 & 2 & -2 \\ 1 & -1 & 5 \\ 2 & -4 & 8 \\ \end{pmatrix} \)

• Result

  The third power: \(\begin{pmatrix} -6 & 14 & -14 \\ 49 & -97 & 161 \\ 56 & -112 & 176 \\ \end{pmatrix}\), the square root: \(\begin{pmatrix} 2-\sqrt{2} & 2\sqrt{2}-2 & 2-2\sqrt{2} \\ 3-2\sqrt{2} & 4\sqrt{2}-5 & 7-4\sqrt{2} \\ 2-\sqrt{2} & 2\sqrt{2}-4 & 6-2\sqrt{2} \\ \end{pmatrix}\)

• Variant 3

  \( \begin{pmatrix} 2 & 0 & 0 \\ -4 & 1 & 3 \\ -4 & 0 & 4 \\ \end{pmatrix} \)

• Result

  The third power: \(\begin{pmatrix} 8 & 0 & 0 \\ -112 & 1 & 63 \\ -112 & 0 & 64 \\ \end{pmatrix}\), the square root: \(\begin{pmatrix} \sqrt{2} & 0 & 0 \\ 2\sqrt{2}-4 & 1 & 1 \\ 2\sqrt{2}-4 & 0 & 2 \\ \end{pmatrix}\)

• Variant 4

  \( \begin{pmatrix} 4 & -2 & 0 \\ 0 & 2 & 0 \\ 6 & -5 & 1 \\ \end{pmatrix} \)

• Result

  The third power: \(\begin{pmatrix} 64 & -56 & 0 \\ 0 & 8 & 0 \\ 126 & -119 & 1 \\ \end{pmatrix}\) the square root: \(\begin{pmatrix} 2 & \sqrt{2}-2 & 0 \\ 0 & \sqrt{2} & 0 \\ 2 & \sqrt{2}-3 & 1 \\ \end{pmatrix}\).