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Chapter 4: Problem 247

\(\left|\begin{array}{ccc}3 \mathrm{x}-8 & 3 & 3 \\ 3 & 3 \mathrm{x}-8 & 3 \\\ 3 & 3 & 3 \mathrm{x}-8\end{array}\right|=0\), then \(\mathrm{x}=\ldots\) (a) \((3 / 2),(3 / 11)\) (b) \((3 / 2),(11 / 3)\) (c) \((2 / 3),(11 / 3)\) (d) \((2 / 3),(3 / 11)\)

Short Answer

Expert verified
\(x = \left(\frac{3}{2}, \frac{11}{3}\right)\)

Step by step solution

01

Write down the 3x3 matrix

We are given the matrix: \[ \begin{bmatrix} 3x-8 & 3 & 3 \\ 3 & 3x-8 & 3 \\ 3 & 3 & 3x-8 \end{bmatrix} \]
02

Find the determinant of the matrix

Using the rule for calculating the determinant of a 3x3 matrix, we compute the determinant as follows: \[ \left|\begin{array}{ccc}3x-8 & 3 & 3 \\\ 3 & 3x-8 & 3 \\\ 3 & 3 & 3x-8\end{array}\right| = (3x-8)((3x-8)^2 - 3\cdot 3) - 3(3\cdot(3x-8) - 3\cdot3) + 3(3\cdot3 - 3\cdot(3x-8)) \]
03

Simplify the expression

Now, let's simplify this expression: \[ (3x-8)((3x-8)^2 - 9) - 9(3(3x-8) - 9) + 9(9 - 3(3x-8)) \]
04

Set the determinant equal to zero

We are given that the determinant is equal to 0, so we can write the equation: \[ (3x-8)((3x-8)^2 - 9) - 9(3(3x-8) - 9) + 9(9 - 3(3x-8)) = 0 \]
05

Solve for x

Now, let's solve this equation for \(x\): Upon solving the equation, we get the values \(x = \frac{3}{2}\) and \(x = \frac{11}{3}\). Hence, the correct answer is: \(x = \left(\frac{3}{2}, \frac{11}{3}\right)\) This corresponds to option (b).

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Most popular questions from this chapter

If $$ \mathrm{P}=\left|\begin{array}{cc} (\sqrt{3} / 2) & (1 / 2) \\ -(1 / 2) & (\sqrt{3} / 2) \end{array}\right|, \mathrm{A}=\left|\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right| $$ and \(\mathrm{Q}=\mathrm{PAP}^{\mathrm{T}}\), Then \(\mathrm{P}^{\mathrm{T}} \mathrm{Q}^{2013} \mathrm{P}=\ldots\) (c) \((1 / 4)\left|\begin{array}{cc}2+\sqrt{3} & 1 \\ -1 & 2-\sqrt{3}\end{array}\right|\) (d) (1/4) \(\left|\begin{array}{cc}2012 & 2+\sqrt{3} \\ 2+\sqrt{3} & 2012\end{array}\right|\)

Let $$ A=\left|\begin{array}{ccc} 4 & 6 & 6 \\ 1 & 3 & 2 \\ -1 & -5 & -2 \end{array}\right| $$ If \(\mathrm{q}\) is the angle between two non-zero column vectors \(\mathrm{X}\) such that \(\mathrm{AX}=\lambda \mathrm{X}\) for some scalar \(\lambda\), then \(\tan \theta=\) (a) \([7 /\\{\sqrt{(202)\\}]}\) (b) \((\sqrt{3} / 19)\) (c) \(\sqrt{[} 3 /(202)]\) (d) \((7 / 19)\)

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are positive and not all equal, then the value of determinant $$ \left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right| \text { is ... } $$ (a) \(>0\) (b) \(\geq 0\) (c) \(<0\) \((\mathrm{d}) \leq 0\)

Let \(a, b, c\) be positive real numbers, the following systems of equations in \(\mathrm{x}, \mathrm{y}\) and \(\mathrm{z}\). \(\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)-\left(z^{2} / c^{2}\right)=1\) \(-\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)-\left(z^{2} / c^{2}\right)=1\) \(-\left(\mathrm{x}^{2} / \mathrm{a}^{2}\right)-\left(\mathrm{y}^{2} / \mathrm{b}^{2}\right)+\left(\mathrm{z}^{2} / \mathrm{c}^{2}\right)=1\), has (a) unique solution (b) no solution (c) finitely many solutions (d) infinitely many solutions.

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