Simplify the Matrix [[x,1,1],[3,-4,4],[-1,-5,2]]=5 , x

Problem

|[x,1,1],[3,−4,4],[−1,−5,2]|=5

Solution

1. Identify the task as solving for x by calculating the determinant of the 3×3 matrix and setting it equal to 5

2. Expand the determinant along the first row using the cofactor expansion method.

x*|[−4,4],[−5,2]|−1*|[3,4],[−1,2]|+1*|[3,−4],[−1,−5]|=5

3. Calculate the 2×2 determinants using the formula a*d−b*c

x*((−4)*(2)−(4)*(−5))−1*((3)*(2)−(4)*(−1))+1*((3)*(−5)−(−4)*(−1))=5

4. Simplify the arithmetic inside the parentheses.

x*(−8+20)−1*(6+4)+1*(−15−4)=5

12*x−10−19=5

5. Combine constants on the left side of the equation.

12*x−29=5

6. Isolate the variable x by adding 29 to both sides and then dividing by 12

12*x=34

x=34/12

7. Reduce the fraction to its simplest form.

x=17/6

Final Answer

x=17/6

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