Multiply the 2x2 Identity Matrix by a variable. The identity matrix is a matrix that when multiplied another matrix will yield the second matrix unaltered. A 2x2 Identity Matrix is as follows:
| 1 0 |
| 0 1 |
multiplying a variable, for this example x, will yield
| x 0 |
| 0 x |
Subtract the altered Identity Matrix from the matrix from which you are seeking the eigenvalues. For this example, the matrix is
| 2 1 |
| 6 1 |
so subtracting the altered Identity Matrix would yield
| 2 1 | | x 0 | | 2-x 1 |
| 6 1 | - | 0 x | = | 6 1-x|
Cross-multiply the numbers of the matrix. Subtract the product of the first and fourth terms by the product of the second and third terms to obtain the matrix's characteristic equation. For the example, cross-multiplying the numbers and subtracting the terms yields
( 2-x ) ( 1-x) - 6 = x^2 - 3x - 4
Make the equation equal 0 and solve for x. For the example,
x^2 -3x - 4 = 0 equals
( x+1 ) ( x-4) = 0 and x = -1 and 4. So the eigenvalues of the matrix are -1 and 4.