Simplify the Matrix

Problem

[[cos(x),−sin(x)],[sin(x),cos(x)]]*[[cos(x),sin(x)],[−sin(x),cos(x)]]

Solution

1. Identify the matrix multiplication rule for two 2×2 matrices, where the element in row i and column j is the dot product of the ith row of the first matrix and the jth column of the second matrix.

2. Calculate the top-left element by multiplying the first row of the first matrix by the first column of the second matrix.

cos(x)*cos(x)+(−sin(x))*(−sin(x))=cos2(x)+sin2(x)

3. Calculate the top-right element by multiplying the first row of the first matrix by the second column of the second matrix.

cos(x)*sin(x)+(−sin(x))*cos(x)=cos(x)*sin(x)−sin(x)*cos(x)

4. Calculate the bottom-left element by multiplying the second row of the first matrix by the first column of the second matrix.

sin(x)*cos(x)+cos(x)*(−sin(x))=sin(x)*cos(x)−cos(x)*sin(x)

5. Calculate the bottom-right element by multiplying the second row of the first matrix by the second column of the second matrix.

sin(x)*sin(x)+cos(x)*cos(x)=sin2(x)+cos2(x)

6. Apply the Pythagorean identity sin2(x)+cos2(x)=1 and simplify the terms that subtract to zero.

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

Final Answer

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

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