Find the Variables

Problem

[[x,y],[−y,x]]−[[y,x],[x,−y]]=[[−1,1],[−3,3]]

Solution

1. Subtract the matrices on the left side by subtracting corresponding entries.

[[x−y,y−x],[−y−x,x−(−y)]]=[[−1,1],[−3,3]]

2. Simplify the entries in the resulting matrix.

[[x−y,y−x],[−x−y,x+y]]=[[−1,1],[−3,3]]

3. Set up a system of equations by equating corresponding entries from both sides.

x−y=−1

y−x=1

−x−y=−3

x+y=3

4. Solve the system using the equations x−y=−1 and x+y=3 Adding these two equations eliminates y

(x−y)+(x+y)=−1+3

2*x=2

x=1

5. Substitute the value of x back into the equation x+y=3 to find y

1+y=3

y=2

6. Verify the solution with the remaining equations.

y−x=2−1=1

−x−y=−1−2=−3

Final Answer

x=1,y=2

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