Find the Inverse [[x,x-4,-1],[2,-1,x-3],[1,-2,3]]=5x-3

Problem

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

Solution

1. Identify the task as solving for x by evaluating the determinant of the given 3×3 matrix and setting it equal to the provided expression 5*x−3

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

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

3. Calculate the 2×2 determinants within the expansion.

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

4. Simplify the terms inside the parentheses.

x*(−3+2*x−6)−(x−4)*(6−x+3)−1*(−4+1)=5*x−3

x*(2*x−9)−(x−4)*(9−x)−1*(−3)=5*x−3

5. Distribute the terms to form a polynomial equation.

(2*x2−9*x)−(9*x−x2−36+4*x)+3=5*x−3

2*x2−9*x−(−x2+13*x−36)+3=5*x−3

2*x2−9*x+x2−13*x+36+3=5*x−3

6. Combine like terms on the left side.

3*x2−22*x+39=5*x−3

7. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by moving all terms to one side.

3*x2−27*x+42=0

8. Divide the entire equation by the greatest common divisor, which is 3 to simplify.

x2−9*x+14=0

9. Factor the quadratic equation.

(x−7)*(x−2)=0

10. Solve for x by setting each factor to zero.

x−7=0⇒x=7

x−2=0⇒x=2

Final Answer

|[x,x−4,−1],[2,−1,x−3],[1,−2,3]|=5*x−3⇒x=7,x=2

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