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Chapter 4: Problem 308

If \(A=\left|\begin{array}{cc}1 & \tan x \\ -\tan x & 1\end{array}\right|\) then \(\mathrm{A}^{\mathrm{T}} \mathrm{A}^{-1}=\) (a) \(\left|\begin{array}{ll}-\cos 2 \mathrm{x} & \sin 2 \mathrm{x} \\ -\sin 2 \mathrm{x} & \cos 2 \mathrm{x}\end{array}\right|\) (b) \(\left|\begin{array}{cc}\cos 2 x & -\sin 2 x \\ \sin 2 x & \cos 2 x\end{array}\right|\) (c) \(\left|\begin{array}{ll}\cos 2 x & \sin 2 x \\ \sin 2 x & \cos 2 x\end{array}\right|\) (d) \(\left|\begin{array}{cc}\tan x & 1 \\ -1 & \tan x\end{array}\right|\)

Short Answer

Expert verified
(c) \(\left[\begin{array}{cc}\cos{2x} & \sin{2x} \\ -\sin{2x} & \cos{2x}\end{array}\right]\)

Step by step solution

01

Find the transpose of matrix A

To find the transpose of matrix A, simply swap the rows and columns. If A is the given matrix, then the transpose of A, denoted as A^T, is \[ A^T = \left[\begin{array}{cc}1 & -\tan{x} \\ \tan{x} & 1\end{array}\right] \]
02

Find the determinant of matrix A

The determinant of a 2x2 matrix is calculated using the formula: \[ \det{(A)} = ad - bc \] For the given matrix A, det(A) can be calculated as follows: \[ \det{(A)} = (1)(1) - (\tan{x})(-\tan{x}) = 1 + \tan^2{x} = \sec^2{x} \]
03

Find the inverse of matrix A

To find the inverse of a matrix, we can use the following formula for a 2x2 matrix: \[ A^{-1} = \frac{1}{\det{(A)}} \left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right] \] So using the det(A) we calculated in step 2, we can find the inverse of A as follows: \[ A^{-1} = \frac{1}{\sec^2{x}} \left[\begin{array}{cc}1 & -\tan{x} \\ \tan{x} & 1\end{array}\right] = \left[\begin{array}{cc}\cos^2{x} & -\sin{x}\cos{x} \\ \sin{x}\cos{x} & \cos^2{x}\end{array}\right] \]
04

Multiply the transpose of A with the inverse of A

Now we'll multiply A^T and A^(-1): \[ A^T A^{-1} = \left[\begin{array}{cc}1 & -\tan{x} \\ \tan{x} & 1\end{array}\right] \cdot \left[\begin{array}{cc}\cos^2{x} & -\sin{x}\cos{x} \\ \sin{x}\cos{x} & \cos^2{x}\end{array}\right] = \left[\begin{array}{cc}\cos{2x} & \sin{2x} \\ -\sin{2x} & \cos{2x}\end{array}\right] \]
05

Compare the results with the given options

Looking at our result from step 4, we can see that the matrix we obtained is equal to option (c): \[ \left[\begin{array}{cc}\cos{2x} & \sin{2x} \\ -\sin{2x} & \cos{2x}\end{array}\right] \] So the correct answer is: (c) \(\left[\begin{array}{cc}\cos{2x} & \sin{2x} \\ -\sin{2x} & \cos{2x}\end{array}\right]\)

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Most popular questions from this chapter

If \(\Delta(x)=\left|\begin{array}{ccc}x^{2}-5 x+3 & 2 x-5 & 3 \\ 3 x^{2}+x+4 & 6 x+1 & 9 \\ 7 x^{2}-6 x+9 & 14 x-6 & 21\end{array}\right|=a x^{3}+b x^{3}+c x+d\) then \(\mathrm{d}=\ldots \ldots\) (a) 156 (b) 187 (c) 119 (d) 141

The value of determinant \(\left|\begin{array}{ccc}\cos ^{2}[(\pi / 2)+x] & \cos ^{2}[(3 \pi / 2)+x] & \cos ^{2}[(5 \pi / 2)+x] \\ \cos [(\pi / 2)+x] & \cos [(3 \pi / 2)+x] & \cos [(5 \pi / 2)+x] \\ \cos [(\pi / 2)-x] & \cos [(3 \pi / 2)-x] & \cos [(5 \pi / 2)-x]\end{array}\right|\) is \(\ldots \ldots\) (a) 0 (b) \(\cos ^{2}[3 \mathrm{x}-(9 \pi / 2)]\) (c) \(\sin ^{2}[(3 \pi / 2)+\mathrm{x}]\) (d) \(\cos ^{2}[(15 \pi / 2)-\mathrm{x}]\)

If the equations \(a x+b y+c z=0,4 x+3 y+2 z=0\) \(\mathrm{x}+\mathrm{y}+\mathrm{z}=0\) have non-trivial solution, then \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in... (a) A.P. (b) G.P. (c) Increasing sequence (d) decreasing sequence.

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

\(\mathrm{f}(\mathrm{x})=\left|\begin{array}{ccc}\cos \mathrm{x} & 0 & \sin \mathrm{x} \\ 0 & 1 & 0 \\ -\sin \mathrm{x} & 0 & \cos \mathrm{x}\end{array}\right|, \mathrm{g}(\mathrm{y})=\left|\begin{array}{ccc}\cos \mathrm{y} & -\sin \mathrm{y} & 0 \\ \sin \mathrm{y} & \cos \mathrm{y} & 0 \\ 0 & 0 & 1\end{array}\right|\) (i) \(\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y})=\) (a) \(\mathrm{f}(\mathrm{xy})\) (b) \(\mathrm{f}(\mathrm{x} / \mathrm{y})\) (c) \(\mathrm{f}(\mathrm{x}+\mathrm{y})\) (d) \(\mathrm{f}(\mathrm{x}-\mathrm{y})\) (ii) Which of the following is correct? (a) \([\mathrm{f}(\mathrm{x})]^{-1}=[1 /\\{\mathrm{f}(\mathrm{x})\\}]\) (b) \([\mathrm{f}(\mathrm{x})]^{-1}=-\mathrm{f}(\mathrm{x})\) (c) \([\mathrm{f}(\mathrm{x})]^{-1}=\mathrm{f}(-\mathrm{x})\) (d) \([\mathrm{f}(\mathrm{x})]^{-1}=-\mathrm{f}(-\mathrm{x})\) (iii) \([\mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{y})]^{-1}=\) (a) \(\mathrm{f}\left(\mathrm{x}^{-1}\right) \mathrm{g}\left(\mathrm{y}^{-1}\right)\) (b) \(\mathrm{f}\left(\mathrm{y}^{-1}\right) \mathrm{g}\left(\mathrm{x}^{-1}\right)\) (c) \(\mathrm{f}(-\mathrm{x}) \mathrm{g}(-\mathrm{y})\) (d) \(\mathrm{g}(-\mathrm{y}) \mathrm{f}(-\mathrm{x})\)
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