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Chapter 4: Problem 326

If $$ \mathrm{A}=\left|\begin{array}{cc} 3 & 1 \\ -9 & -3 \end{array}\right| $$ then \(\mathrm{I}+2 \mathrm{~A}+3 \mathrm{~A}^{2}+\ldots \ldots \infty=\ldots\) (a) \(\mid\)\begin{tabular}{cc|c|cc|c|cc|c|cc} 9 & 1 & (b) & 4 & 1 & (c) & \(\begin{gathered}7 & 2 \\ -18 & -5\end{gathered} \mid\) (d) \(\left|\begin{array}{cc}7 & 2 \\ -5 & -18\end{array}\right|\) \end{tabular}

Short Answer

Expert verified
The given matrix series does not converge, as the absolute values of the eigenvalues of matrix A are greater than 1. Therefore, there is no defined sum for the series and no solution.

Step by step solution

01

Calculate eigenvalues of matrix A

To calculate the eigenvalues of matrix A, we use the matrix determinant formula: \[ \det(\mathrm{A}-\lambda\mathrm{I}) = 0 \] Setting up the equation for matrix A, we have: \[ \det\left(\left|\begin{array}{cc} 3-\lambda & 1 \\\ -9 & -3-\lambda \end{array}\right|\right)=0 \] Expanding the determinant, we have: \[ (3-\lambda)(-3-\lambda) - (-9)(1) = 0 \]
02

Solve for eigenvalues

Now we'll solve the quadratic equation for the eigenvalues: \[ \lambda^2 - 9 = 0 \] The eigenvalues are: \[ \lambda_1 = 3, \lambda_2 = -3 \]
03

Check the convergence of the series

The convergence of the series depends on the absolute value of the eigenvalues. The absolute values for the eigenvalues are: \[ |\lambda_1| = 3, |\lambda_2| = 3 \] Since both eigenvalues have an absolute value greater than 1, the given series doesn't converge, and there is no defined sum for the series. As we can see, the series does not converge. Therefore, we can conclude that the given problem has no solution.

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Most popular questions from this chapter

If the system of equations \(x+k y+3 z=0,3 x+k y-2 z=0\) \(2 \mathrm{x}+3 \mathrm{y}-4 \mathrm{z}=0\) has non trivial solution, then \(\left(\mathrm{xy} / \mathrm{z}^{2}\right)=\ldots\) (a) \((5 / 6)\) (b) \(-(5 / 6)\) (c) \((6 / 5)\) (d) \(-(6 / 5)\)

Let \(\lambda\) and \(\alpha\) be real. The system of equations \(\lambda x+(\sin \alpha) y+(\cos \alpha) z=0\) \(x+(\cos \alpha) y+(\sin \alpha) z=0\) \(-x+(\sin \alpha) \mathrm{y}-(\cos \alpha) \mathrm{z}=0\) has no trivial solution. (i) The set of all values of \(\lambda\) is (a) \([-\sqrt{3}, \sqrt{3}]\) (b) \([-\sqrt{2}, \sqrt{2}]\) (c) \([-1,1]\) (d) \([0,(\pi / 2)]\) (ii) For \(\lambda=1, \alpha=\ldots .\) (a) \(\mathrm{n} \pi, \mathrm{n} \pi-(\pi / 4)\) (b) \(2 \mathrm{n} \pi, \mathrm{n} \pi-(\pi / 4)\) (c) \(\mathrm{n} \pi, \mathrm{n} \pi+(\pi / 4)\) (d) \(-\pi,-(3 \pi / 4)\)

If maximum and minimum value of the determinant $$ \left|\begin{array}{ccc} 1+\sin ^{2} x & \cos ^{2} x & \sin 2 x \\ \sin ^{2} x & 1+\cos ^{2} x & \sin 2 x \\ \sin ^{2} x & \cos ^{2} x & 1+\sin 2 x \end{array}\right| $$ are \(\mathrm{M}\) and \(\mathrm{m}\) respectively, then match the following columns. \begin{tabular}{|l|l|} \hline \multicolumn{1}{|c|} { Column I } & \multicolumn{1}{c|} { Column II } \\\ \hline (1) \(\mathrm{M}^{2}+\mathrm{m}^{2013}=\) & (A) always an odd for \(\mathrm{k} \in \mathrm{N}\) \\ (2) \(\mathrm{M}^{3}-\mathrm{m}^{3}=\) & (B) Being three sides of triangle \\ (3) \(\mathrm{M}^{2 \mathrm{k}}-\mathrm{m}^{2 \mathrm{k}}=\) & (C) 10 \\ (4) \(2 \mathrm{M}-3 \mathrm{~m}, \mathrm{M}+\mathrm{m}, \mathrm{M}+2 \mathrm{~m}\) & (D) 4 \\ & (E) Always an even for \(\mathrm{k} \in \mathrm{N}\) \\ & (F) Does not being three sides of triangle. \\ & (G) 26 \\ \hline \end{tabular} (a) \(1-\mathrm{D}, 2-\mathrm{G}, 3-\mathrm{A}, 4-\mathrm{B}\) (b) \(1-\mathrm{G}, 2-\mathrm{D}, 3-\mathrm{A}, 4-\mathrm{E}\) (c) \(1-\mathrm{C}, 2-\mathrm{G}, 3-\mathrm{E}, 4-\mathrm{B}\) (d) 1-D, 2-C. 3-E. 4-F

If \(f(x)=\left|\begin{array}{ccc}2 \cos ^{2} x & \sin 2 x & -\sin x \\ \sin 2 x & 2 \sin ^{2} x & \cos x \\ \sin x & -\cos x & 0\end{array}\right|\) then \((\pi / 2) \int_{0}\left[f(x)+f^{\prime}(x)\right] d x=\ldots .\) (a) 1 (b) \(\pi\) (c) 0 (d) \([(1 / 3)-(\pi / 2)]\)

If $$ A_{r}=\left|\begin{array}{cc} r & r-1 \\ r-1 & r \end{array}\right| $$ where \(\mathrm{r}\) is a natural number than the value of \(\left.\sqrt{[}^{2013} \sum_{r=1} A_{r}\right]\) is (a) 1 (b) 40 (c) 2012 (d) 2013
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