For all values of θwhere, 0<θ<(π)/(2), the determinant of ( 1 -cotθ cotθ 1 )(

step 1 In the problem, late x equal A plus B, where the A and B are given in the problem. Now, A equals

For all values of θwhere, 0<θ<(π)/(2), the determinant of ( ...

For all values of $\theta$ where, $0<\theta<\frac{\pi}{2}$, the determinant of $\left(\begin{array}{cc}1 & -\cot \theta \\ \cot \theta & 1\end{array}\right)\left(\begin{array}{cc}1 & \cot \theta \\ -\cot \theta & 1\end{array}\right)^{-1}+\left(\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right)$ $\left(\begin{array}{cc}1 & \tan \theta \\ -\tan \theta & 1\end{array}\right)^{-1}$ lies in the interval. (a) $(0,4]$ (b) $(-2,2)$ (c) $[-1,1]$ (d) $[1,2]$

For all values of $\theta$ where, $0<\theta<\frac{\pi}{2}$, the determinant of $\left(\begin{array}{cc}1 & -\cot \theta \\ \cot \theta & 1\end{array}\right)\left(\begin{array}{cc}1 & \cot \theta \\ -\cot \theta & 1\end{array}\right)^{-1}+\left(\begin{array}{cc}1 & -\tan \theta \\ \tan \theta & 1\end{array}\right)$
$\left(\begin{array}{cc}1 & \tan \theta \\ -\tan \theta & 1\end{array}\right)^{-1}$ lies in the interval.
(a) $(0,4]$
(b) $(-2,2)$
(c) $[-1,1]$
(d) $[1,2]$

Key Concepts

Interval Analysis and Function Range

Interval analysis involves determining the range or set of values that a function can take as its parameter varies over a specified domain. In the context of the problem, after simplifying the matrix expression and computing the determinant, one must analyze how the value of the determinant changes as the parameter (in this case, ?) varies within the interval (0, ?/2). This step is crucial in matching the computed range to one of the given interval choices.

Trigonometric Functions in Matrices

Trigonometric functions such as tangent and cotangent are used in the entries of the matrices in this problem, and their properties in the interval (0, ?/2) are critical. Understanding the behavior, limits, and identities of these functions is essential for simplifying the matrix expressions. The knowledge of how these functions behave in different intervals helps ensure that operations like inversion are valid and that the determinant’s value can be correctly evaluated over the specified domain.

Matrix Multiplication and Simplification

Matrix multiplication is a fundamental operation in linear algebra that combines matrices to produce another matrix following specific rules. The problem involves products and sums of matrices, including products where inverse matrices are present. Recognizing and using properties such as the distributive law and the relationship between multiplication and inversion are key to simplifying the given expression and ultimately determining the range of its determinant.

Determinant of a Matrix

The determinant is a scalar value that can be computed from the elements of a square matrix, and it provides important information about the matrix such as whether it is invertible. In the context of this problem, the determinant of the given matrix expression is evaluated to study its behavior over a range of parameter values. Key properties include the multiplicative property (the determinant of a product of matrices equals the product of their determinants) and the relation between a matrix and its inverse, which are both used in simplifying complex matrix expressions.

Matrix Inversion

Matrix inversion is the process of finding a matrix, called the inverse, which when multiplied by the original matrix yields the identity matrix. In this problem, the presence of inverse matrices (specifically of matrices with trigonometric entries) requires an understanding of how to compute and utilize the inverse of a 2x2 matrix. The technique often involves using a formula based on swapping and negating entries, and dividing by the determinant, assuming the matrix is nonsingular.

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