# Some Definitions > **Transpose** > The transpose of a matrix is defined by: > $(A^T)_{ij} = A_{ji}$ > **Inverse** > The *inverse* of a square matrix is defined by: > $A^{-1} A = A A^{-1} = I$ > When the inverse exists it is unique. ^b970df # Properties $ \begin{align} (AB)^T &= B^T A^T, \\ (AB)^{-1} &= B^{-1} A^{-1}, \\ \det(AB) &= \det(A) \det(B), \\ \det(B^{-1} A B) &= \det(A), \\ \text{tr}(B^{-1} A B) &= \text{tr}(A),\\ \det(A^T) &= \det(A),\\ {\text{eigenvalues of} \, A^T} &= {\text{eigenvalues of} \, A} , \\ \det(A) &= \lambda_1 \lambda_2 \ldots \lambda_n \\ \text{tr}(A) &= \lambda_1 +\lambda_2 +\ldots +\lambda_n \end{align} $