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Eigenvalues of sum of anti-commuting matrices

I know two anti-commuting (nxn)-matrices A and B, n -even. I know also that +-a and +-b are real eigenvalues among all eigenvalues of A and B respectively. How to show that the matrix A+B has also at least two real eigenvalues of the form +-\sqrt{a^2+b^2} ? (I know also that (A+B)(A+B)^t=(A+B)^t(A+B)=(a^2+b^2)I, where I is (nxn)-identity matrix and AA^t=A^tA=a^2I, BB^t=B^tB=b^2I and ^t denotes transposition).