If I=`[{:(,1,0),(,0,1):}], J=[{:(,0,1),(,-1,0):}]and B=[{:(,cos theta,sin theta),(,-sin theta,cos theta)]:}"then B"=`

Options

A. `Icos theta+J sin theta`
B. `Icostheta-Jsintheta`
C. `I sin theta+Jcos theta`
D. `-Icos theta+Jsin theta`

Answer

To solve the problem, we need to express the matrix \( B \) in terms of the identity matrix \( I \) and the matrix \( J \). Let's go through the steps systematically.

### Step 1: Define the matrices
We have the following matrices defined:
- Identity matrix \( I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)
- Matrix \( J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \)
- Matrix \( B = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \)

### Step 2: Express \( B \) in terms of \( I \) and \( J \)
We want to express \( B \) in the form:
\[ B = aI + bJ \]
where \( a \) and \( b \) are scalars.

### Step 3: Write the expression for \( aI + bJ \)
Substituting the matrices into the expression, we have:
\[ aI + bJ = a \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + b \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]
This simplifies to:
\[ = \begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix} + \begin{pmatrix} 0 & b \\ -b & 0 \end{pmatrix} = \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \]

### Step 4: Set the matrices equal to each other
Now we set this equal to matrix \( B \):
\[ \begin{pmatrix} a & b \\ -b & a \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \]

### Step 5: Equate corresponding elements
From the equality of the matrices, we can derive the following equations:
1. \( a = \cos \theta \)
2. \( b = \sin \theta \)
3. \( -b = -\sin \theta \) (which is consistent with \( b = \sin \theta \))
4. \( a = \cos \theta \) (which is consistent with the first equation)

### Step 6: Substitute back to find \( B \)
Now substituting the values of \( a \) and \( b \) back into the expression for \( B \):
\[ B = aI + bJ = \cos \theta I + \sin \theta J \]
Thus, we can express \( B \) as:
\[ B = \cos \theta I + \sin \theta J \]

### Conclusion
The correct representation of matrix \( B \) is:
\[ B = I \cdot \cos \theta + J \cdot \sin \theta \]

### Final Answer
The correct option is:
1. \( B = I \cdot \cos \theta + J \cdot \sin \theta \)

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