Find the Inverse [[cos(0),-sin(0)],[sin(0),cos(0)]]

Problem

[[cos(0),−sin(0)],[sin(0),cos(0)]]

Solution

1. Evaluate the trigonometric functions at 0 Since cos(0)=1 and sin(0)=0 the matrix simplifies to the identity matrix.

[[1,0],[0,1]]

2. Identify the properties of the identity matrix I The inverse of the identity matrix is the identity matrix itself.

I(−1)=I

3. Verify using the formula for the inverse of a 2×2 matrix [[a,b],[c,d]](−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

det(=)*(1)*(1)−(0)*(0)=1

4. Apply the formula to find the inverse.

1/1*[[1,0],[0,1]]=[[1,0],[0,1]]

Final Answer

[[cos(0),−sin(0)],[sin(0),cos(0)]](−1)=[[1,0],[0,1]]

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