Find Reduced Row Echelon Form [[cos(1),sin(1)],[-sin(1),cos(1)]]

Problem

[[cos(1),sin(1)],[−sin(1),cos(1)]]

Solution

1. Identify the matrix properties. The given matrix is a rotation matrix of the form R=[[cos(θ),sin(θ)],[−sin(θ),cos(θ)]] where θ=1 radian.

2. Calculate the determinant of the matrix to determine if it is invertible.

det(A)=(cos(1))*(cos(1))−(sin(1))*(−sin(1))

det(A)=cos2(1)+sin2(1)

det(A)=1

3. Conclude that since the determinant is non-zero (det(A)=1≠0, the matrix is non-singular (invertible).

4. Determine the Reduced Row Echelon Form (RREF). Any invertible n×n matrix can be reduced to the n×n identity matrix (I_n) through elementary row operations.

Final Answer

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

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