Question

If A and B are two matrices such that AB=B and BA=A, $\left( {{A}^{2}}+{{B}^{2}} \right)=\lambda \left( A+B \right)$. Considering $f\left( x \right)=\left| \left[ \sin x \right]+\left[ \cos x \right] \right|$; where $\left[ \text{ } \right ]$ is the greatest integer function. Then find $f\left( 4\lambda \right)$?

Answer 1

For this we have to calculate the value of $f\left( 4\lambda \right)$ where we have the definition of the function $f\left( x \right)$ and a relation to get the value of $\lambda$. In the problem we have given the relations between the two matrices $A$ and $B$ as $AB=B$ and $BA=A$, the equation which is having $\lambda$ value is $\left( {{A}^{2}}+{{B}^{2}} \right)=\lambda \left( A+B \right)$. Now we will try to calculate the value of ${{A}^{2}}$, ${{B}^{2}}$ from the relations we have and here we will use the associative property of the matrix multiplication and calculate the values. After getting the values of ${{A}^{2}}$, ${{B}^{2}}$ we will use the equation $\left( {{A}^{2}}+{{B}^{2}} \right)=\lambda \left( A+B \right)$ to get the $\lambda$ value. After getting the $\lambda$ value we can simply calculate the required value which is $f\left( 4\lambda \right)$.

Complete step-by-step solution:
Given that, $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$.
Now the value of ${{A}^{2}}$ can be calculated as
$\Rightarrow {{A}^{2}}=A.A$
We have the relation $BA=A$. Substituting this value in the above equation, then we will get
$\Rightarrow {{A}^{2}}=A\left( BA \right)$
From the associative law of matrix multiplication, we can write $A\left( BA \right)=\left( AB \right)A$ in the above equation, then we will have
$\Rightarrow {{A}^{2}}=\left( AB \right)A$
Again, we have the relation $AB=B$. From this relation the above equation is modified as
$\begin{aligned} & \Rightarrow {{A}^{2}}=BA \\\ & \Rightarrow {{A}^{2}}=A....\left( \text{i} \right) \\\ \end{aligned}$
Now the value of ${{B}^{2}}$ can be calculated as
$\Rightarrow {{B}^{2}}=B.B$
Substituting the value $AB=B$ in the above equation, then we will get
$\Rightarrow {{B}^{2}}=B\left( AB \right)$
Applying the associative law of matrix multiplication in the above equation, then we will have
$\Rightarrow {{B}^{2}}=\left( BA \right)B$
Again, using the relation $BA=A$ in the above equation, then we will get
$\begin{aligned} & \Rightarrow {{B}^{2}}=AB \\\ & \Rightarrow {{B}^{2}}=B.....\left( \text{ii} \right) \\\ \end{aligned}$
From equations $\left( \text{i} \right)$ and $\left( \text{ii} \right)$ we can write the value of ${{A}^{2}}+{{B}^{2}}$ is given by
$\Rightarrow {{A}^{2}}+{{B}^{2}}=A+B$
Comparing the above equation with the given equation which is $\left( {{A}^{2}}+{{B}^{2}} \right)=\lambda \left( A+B \right)$, then we will get
$\Rightarrow \lambda =1$.
Now from the function $f\left( x \right)=\left| \left[ \sin x \right]+\left[ \cos x \right] \right|$ where $\left[ {} \right]$ is the greatest integer function. The value of $f\left( 4\lambda \right)$ is given by
$\begin{aligned} & \Rightarrow f\left( 4\lambda \right)=\left| \left[ \sin \left( 4\times 1 \right) \right]+\left[ \cos \left( 4\times 1 \right) \right] \right| \\\ & \Rightarrow f\left( 4\lambda \right)=\left| \left[ \sin 4 \right]+\left[ \cos 4 \right] \right| \\\ \end{aligned}$
In the above equation we have the values $\sin 4$, $\cos 4$. We know that the value $4$ lies between $\pi$, $\dfrac{3\pi }{2}$ i.e., the angle lies in the third Quadrant. In the third quadrant the maximum value of $\sin x$, $\cos x$ is $-1$. Then we will get the value of $f\left( 4\lambda \right)$ will be
$\begin{aligned} & \Rightarrow f\left( 4\lambda \right)=\left| -1-1 \right| \\\ & \Rightarrow f\left( 4\lambda \right)=\left| -2 \right| \\\ & \therefore f\left( 4\lambda \right)=2 \\\ \end{aligned}$

Note: In this problem we have the term greatest integer function. It is the function which gives the greatest value of a given expression for the given range. When we have this function we have written the maximum values of functions $\sin x$, $\cos x$ in the quadrant in which the calculated angle belongs to. Students may think that the value $\sin 4$ as $\sin 4{}^\circ$. Don’t do that because there is a lot of difference in the values of $\sin 4$, $\sin 4{}^\circ$. So please keep the value as an integer and don’t assume it as a degree.