Question:
If $A + B + C = 180^\circ$, then the value of $\begin{bmatrix}
1& 1 & 0 \\[0.3em]
-\cos\,C & \cot\, A & -\cot\, A \\[0.3em]
-\cot\,B & \cot\,C & \cot\, B \end{bmatrix}$ is
If $A + B + C = 180^\circ$, then the value of $\begin{bmatrix}
1& 1 & 0 \\[0.3em]
-\cos\,C & \cot\, A & -\cot\, A \\[0.3em]
-\cot\,B & \cot\,C & \cot\, B \end{bmatrix}$ is
Updated On: Jul 6, 2022
- $1 + \cot A \cot B$
- $1 + \cot B \cot C $
- $1 + \cot C \cot A$
- none of these.
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The Correct Option is A
Solution and Explanation
$\begin{vmatrix}
1& 1 & 0 \\[0.3em]
-\cos\,C & \cot\, A & -\cot\, A \\[0.3em]
-\cot\,B & \cot\,C & \cot\, B \end{vmatrix}$
= $\begin{vmatrix}
1& 1 & 0 \\[0.3em]
-\cos\,C & \cot\, A +\cot \, C & -\cot\, A \\[0.3em]
-\cot\,B & \cot \, B + \cot\,C & \cot\, B \end{vmatrix}$
= $ \cot A \cot B + \cot B \cot C + \cot A \cot B + \cot A \cot C$
= $ \cot A \cot B + (\cot A \cot B + \cot B \cot C + \cot C \cot A)$
Since $A + B + C = 108^\circ$
$\therefore\: \tan (A + B) = \tan (180^\circ - C) = - \tan C $
$\Rightarrow$ $\frac{\tan \,A \,\tan \,B }{1 - \tan \, A \, \tan \, B} = - \, \tan \, C $
$\Rightarrow \: \tan A + \tan B + \tan C = \tan A \tan B \tan C $
$\Rightarrow\:\: \frac{\tan \, A}{\tan\, A\, \tan\, B\, \tan \,C } + \frac{\tan \, B}{\tan\, A\, \tan\, B\, \tan \,C } +\frac {\tan \,C}{\tan\, A\, \tan\, B\, \tan \,C } = 1 $
$\Rightarrow\: \cot B \cot C + \cot C \cot A + \cot A \cot B = 1$
$\therefore$ given value = $1 + \cot A \cot B$
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.