Multiply the Matrices

Problem

[[cos(45),0,sin(45)],[0,1,0],[−sin(45),0,cos(45)]]*[[√(,2),0,0],[0,√(,2),0],[0,0,√(,2)]]

Solution

1. Evaluate the trigonometric functions for 45 Since cos(45)=√(,2)/2 and sin(45)=√(,2)/2 the first matrix becomes:

[[√(,2)/2,0,√(,2)/2],[0,1,0],[−√(,2)/2,0,√(,2)/2]]

2. Identify that the second matrix is a scalar matrix, which is equivalent to multiplying the first matrix by the scalar √(,2)

√(,2)*[[√(,2)/2,0,√(,2)/2],[0,1,0],[−√(,2)/2,0,√(,2)/2]]

3. Distribute the scalar √(,2) to every element in the first matrix.

[[√(,2)⋅√(,2)/2,√(,2)⋅0,√(,2)⋅√(,2)/2],[√(,2)⋅0,√(,2)⋅1,√(,2)⋅0],[√(,2)⋅(−√(,2)/2),√(,2)⋅0,√(,2)⋅√(,2)/2]]

4. Simplify the products using the identity √(,2)⋅√(,2)=2

[[2/2,0,2/2],[0,√(,2),0],[−2/2,0,2/2]]

5. Reduce the fractions to find the final matrix elements.

[[1,0,1],[0,√(,2),0],[−1,0,1]]

Final Answer

[[cos(45),0,sin(45)],[0,1,0],[−sin(45),0,cos(45)]]*[[√(,2),0,0],[0,√(,2),0],[0,0,√(,2)]]=[[1,0,1],[0,√(,2),0],[−1,0,1]]

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