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Chapter 4: Problem 317

The set of natural numbers \(\mathrm{N}\) is partitioned into arrays of rows and columns in the form of matrices as \(\mathrm{M}_{1}=|1|, \mathrm{M}_{2}=\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|, \mathrm{M}_{3}=\left|\begin{array}{ccc}6 & 7 & 8 \\ 9 & 10 & 11 \\ 12 & 13 & 14\end{array}\right|, \ldots, \mathrm{M}_{\mathrm{n}}=[\ldots]\) and so on. Find the sum of elements of the diagonal in \(\mathrm{M}_{6}\). (a) 144 (b) 441 (c) 321 (d) 461

Short Answer

Expert verified
The sum of the diagonal elements of M6 is \(64\).

Step by step solution

01

Generate the M6 matrix

To generate the M6 matrix, we can observe that each matrix in the sequence contains all natural numbers between the sums of the first natural numbers. For example, in M2, the elements are 2 to 5, which are between 1+1 = 2 and 1+2+3 = 6, and in M3, the elements are from 6 to 14, which are between 1+2+3 = 6 and 1+2+3+4 = 10. Based on the pattern, we can determine the range of elements in M6: 1+2+3+4+5 = 15 and 1+2+3+4+5+6+7 = 28. Therefore, the elements in M6 are from 15 to 28. The matrix M6 is formed as follows: \(\mathrm{M}_{6}=\left|\begin{array}{cccccc}15 & 16 & 17 & 18 & 19 & 20 \\\ 21 & 22 & 23 & 24 & 25 & 26 \\\ 27 & 28 & - & - & - & - \end{array}\right|\)
02

Identify the diagonal elements in M6

The diagonal elements of M6 are the elements in the matrix that are located on a line from the top-left corner to the bottom-right corner. In this case, the diagonal elements are 15, 22, and 27.
03

Calculate the sum of the diagonal elements

Now, we can calculate the sum of the diagonal elements in M6: 15 + 22 + 27 = 64. Although the provided answer options do not include 64, the sum of the diagonal elements of M6 is 64.

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Most popular questions from this chapter

When the determinant $$ f(x)=\left|\begin{array}{ccc} \cos 2 x & \sin ^{2} x & \cos 4 x \\ \sin ^{2} x & \cos 2 x & \cos ^{2} x \\ \cos 4 x & \cos ^{2} x & \cos 2 x \end{array}\right| $$ is expanded in powers of \(\sin \mathrm{x}\), then the constant term in that expansion is (a) 0 (b) \(-1\) (c) 2 (d) 1

If \(\mathrm{A}=\left|\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & \mathrm{a} & 1\end{array}\right|, \mathrm{A}^{-1}=\left|\begin{array}{ccc}(1 / 2) & -(1 / 2) & (1 / 2) \\ -4 & 3 & \mathrm{c} \\ (5 / 2) & -(3 / 2) & (1 / 2)\end{array}\right|\) then (a) \(\mathrm{a}=2, \mathrm{c}=-(1 / 2)\) (b) \(\mathrm{a}=1, \mathrm{c}=-1\) (c) \(\mathrm{a}=-1, \mathrm{c}=1\) (d) \(\mathrm{a}=(1 / 2), \mathrm{c}=(1 / 2)\)

Suppose a matrix A satisfies \(\mathrm{A}^{2}-5 \mathrm{~A}+7 \mathrm{I}=0 .\) If \(\mathrm{A}^{5}=\mathrm{aA}+\mathrm{bI}\) then the value of \(2 \mathrm{a}-3 \mathrm{~b}\) must be (a) 4135 (b) 1435 (c) 1453 (d) 3145

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

\(\left|\begin{array}{ccc}\sin (x+p) & \sin (x+q) & \sin (x+r) \\ \sin (y+p) & \sin (y+q) & \sin (y+r) \\ \sin (z+p) & \sin (z+q) & \sin (z+r)\end{array}\right|=\ldots\) (a) \(\sin (x+y+z)\) (b) \(\sin (\mathrm{p}+\mathrm{q}+\mathrm{r})\) (c) 1 (d) 0
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