Question

Evaluate the value of the matrix: \[\Delta =\left| \begin{matrix}
   \cos \alpha \cos \beta & \cos \alpha \sin \beta & -\sin \alpha \\
   -\sin \beta & \cos \beta & 0 \\
   \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha \\
\end{matrix} \right|\].

Answer 1

Hint: Use the fact that the value of a $3\times 3$ determinant of the form $\left| \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right|$ is $a\left( ei-fh \right)-b\left( di-gf \right)+c(dh-eg)$. Substitute the value of the variables by comparing it with the given matrix. Simplify the value of the matrix using the trigonometric identity ${{\cos }^{2}}x+{{\sin }^{2}}x=1$.

Step-by-step answer:
We have to calculate the value of the matrix \[\Delta =\left| \begin{matrix}
   \cos \alpha \cos \beta & \cos \alpha \sin \beta & -\sin \alpha \\
   -\sin \beta & \cos \beta & 0 \\
   \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha \\
\end{matrix} \right|\]. We observe that this matrix has 3 rows and 3columns. Thus, it is a $3\times 3$ matrix.
We will now calculate the value of the given matrix using the fact that the value of the matrix of the form $\left| \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right|$ is $a\left( ei-fh \right)-b\left( di-gf \right)+c(dh-eg)$.
We will now substitute the values $a=\cos \alpha \cos \beta ,b=\cos \alpha \sin \beta ,c=-\sin \alpha $, $d=-\sin \beta ,e=\cos \beta ,f=0$ and $g=\sin \alpha \cos \beta ,h=\sin \alpha \sin \beta ,i=\cos \alpha $ in each row of the matrix.
Thus, we have $\Delta =\cos \alpha \cos \beta \left( \cos \alpha \cos \beta -0\left( \cos \alpha \cos \beta \right) \right)-\cos \alpha \sin \beta \left( -\cos \alpha \sin \beta -0\left( \sin \alpha {{\cos }^{2}}\beta \right) \right)-\sin \alpha \left( -\sin \alpha {{\sin }^{2}}\beta -\sin \alpha {{\cos }^{2}}\beta \right)$.
Simplifying the above expression, we have $\Delta ={{\cos }^{2}}\alpha {{\cos }^{2}}\beta +{{\cos }^{2}}\alpha {{\sin }^{2}}\beta +{{\sin }^{2}}\alpha {{\sin }^{2}}\beta +{{\sin }^{2}}\alpha {{\cos }^{2}}\beta $.
Rearranging the terms of the above equation, we have $\Delta ={{\cos }^{2}}\alpha \left( {{\cos }^{2}}\beta +{{\sin }^{2}}\beta \right)+{{\sin }^{2}}\alpha \left( {{\cos }^{2}}\beta +{{\sin }^{2}}\beta \right)$.
We know the trigonometric identity ${{\cos }^{2}}x+{{\sin }^{2}}x=1$.
Thus, we have $\Delta ={{\cos }^{2}}\alpha \left( 1 \right)+{{\sin }^{2}}\alpha \left( 1 \right)={{\cos }^{2}}\alpha +{{\sin }^{2}}\alpha $.
So, we have $\Delta ={{\cos }^{2}}\alpha +{{\sin }^{2}}\alpha =1$.
Hence, the value of the matrix \[\Delta =\left| \begin{matrix}
   \cos \alpha \cos \beta & \cos \alpha \sin \beta & -\sin \alpha \\
   -\sin \beta & \cos \beta & 0 \\
   \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha \\
\end{matrix} \right|\] is 1.

Note: We can expand the matrix along any row or column. However, in each case, the value of the matrix remains the same. A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. The dimension of a matrix is written as the product of the number of rows and columns. Two matrices can be added or subtracted element by element if they have the same dimension.