Solutions to a Class of Equations in Integral Matrices of Order 2

To solve problems of integer matrix equations related to Pythagorean equation x²+y²=z², the solutions ( X , Y ) to the 2×2 integral matrix equation ${\mathit{\boldsymbol{X}}^2} + {\mathit{\boldsymbol{Y}}^2} = \lambda \mathit{\boldsymbol{I}}$, where $\lambda \in \mathbb{Z}$ and I is the unit matrix, which are related to the Pythagorean equation, are investigated and completely solved by using the basic operation of matrix to transform the problem of integer matrix equation into the problem of solving some Diophantine equations, which is gradually extended from the special case to the general case. The solutions to 2×2 integral matrix equation ${\mathit{\boldsymbol{X}}^2} - {\mathit{\boldsymbol{Y}}^2} = \lambda \mathit{\boldsymbol{I}}$ also can be solved with similar methods.