This means that there is no real number (the kind of number that we learned when we were little kids) that satisfies this formula. Quadratic and cubic equations sometimes have no real solutions. And such things are not so easy to calculate.įortunately, we have the eigenvalue and eigenvector calculator that can hide all these ugly formulas and effortlessly give us a pretty answer. However, when the arrays are of size 3x3, we obtain a cubic equation, i.e., an equation with the variable to the third power. In the case of 2x2 matrices, it all boils down to the quadratic formula. In particular, the formulas from above don't work here. However, the trick is that this time the equation is far more complicated. Where the multiplication on the left is matrix multiplication. Now, when it comes to how to find eigenvectors and eigenvalues, the definition is again the same: they are the numbers λ and vectors v that satisfy the matrix equation: However, the determinant is now a more complicated manner: For instance, the trace is the sum of the cells on the main diagonal: In general, most of the definitions above are the same for 3x3 matrices. Where for us, the entries a₁, a₂, up to c₃ are real numbers.
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