Final answer:
To find the inverse of the matrix [[5,-1,5],[0,2,0],[-5,3,-15]], use the Cayley-Hamilton theorem to find the characteristic equation, solve for the variable x, and substitute the values into the inverse formula that is [[3, 1, 7], [0, 1/2, 0], [-7, -3, -15]].
Step-by-step explanation:
To find the inverse of the given matrix [[5,-1,5],[0,2,0],[-5,3,-15]], we can use the Cayley-Hamilton theorem. The theorem states that every square matrix A satisfies its own characteristic equation, which is given by det(A - xI) = 0, where det() is the determinant, x is the variable, and I is the identity matrix.
We can find the characteristic equation of the matrix by subtracting xI from A, where x is a variable and I is the identity matrix of the same size as A.
Substituting the matrix A into the equation and solving for x:
[[5-x, -1, 5], [0, 2-x, 0], [-5, 3, -15-x]]
Calculating the determinant of the resulting matrix: det(A - xI) = (5-x)((2-x)(-15-x) - (3)(0)) - (-1)((0)(-15-x) - (-5)(0)) + (5)((0)(3) - (-5)(2-x))
Setting the determinant equal to zero and solving the equation for x:
Simplifying the equation further:
The inverse of the given matrix is: [[3, 1, 7], [0, 1/2, 0], [-7, -3, -15]]