Multiply the identity matrix by a variable. An identity matrix is a matrix that has 1s along its diagonal from the upper left-hand corner to the lower right-hand corner and 0s everywhere else. The identity matrix's function is that it can be multiplied to a matrix and will result in that matrix as an answer. For example, for 2x2 matrices, the first matrix has the values 1 and 2 in its top rows, and 2 and 0 in its bottom row. The product of multiplying an identity matrix to this matrix will result in having the same values in the same positions: 1 and 2 will be in the top row still, and 2 and 0 will be in the bottom row. Any variable will suffice since this is to just give structure to the polynomial.
Subtract the original matrix from the identity matrix. For example, if x was chosen for the variable, then the identity matrix will read x and 0 in the top row and 0 and x in the bottom row. Subtracting the example matrix of 1 and 2 in the top row and 2 and 0 in the bottom row will yield the terms x-1 and -2 in the top row and -2 and x in the bottom row.
Find the determinate of the new matrix. For a 2x2 matrix, find the product of the first and fourth terms and the product of the second and third terms. Subtract the second/third product from the first/fourth one. Multiplying the first term of x-1 by the fourth term of x will yield the term x2 (x squared)-x and multiplying the second term of -2 by the third term of -2 will result in 4. Subtract 4 from x2-x.
Organize the terms from greatest power to least. The equation will read " x2-x-4." This is the characteristic polynomial.