Multiply the Matrices

Problem

[[cos(45),−sin(45)],[sin(45),cos(45)]]*[[2],[−3]]

Solution

1. Evaluate the trigonometric functions for 45

cos(45)=√(,2)/2

sin(45)=√(,2)/2

2. Substitute these values back into the matrix.

[[√(,2)/2,−√(,2)/2],[√(,2)/2,√(,2)/2]]*[[2],[−3]]

3. Apply the matrix multiplication rule for a 2×2 matrix and a 2×1 vector.

[[(√(,2)/2)*(2)+(−√(,2)/2)*(−3)],[(√(,2)/2)*(2)+(√(,2)/2)*(−3)]]

4. Simplify the arithmetic expressions within the vector.

[[√(,2)+(3√(,2))/2],[√(,2)−(3√(,2))/2]]

5. Combine the terms using a common denominator.

[[(2√(,2)+3√(,2))/2],[(2√(,2)−3√(,2))/2]]

[[(5√(,2))/2],[−√(,2)/2]]

Final Answer

[[cos(45),−sin(45)],[sin(45),cos(45)]]*[[2],[−3]]=[[(5√(,2))/2],[−√(,2)/2]]

Want more problems? Check here!