Category Archives: pow 行列式

以伴隨矩陣的行列式表達分塊矩陣的行列式。 Suppose is , is , is , and is a number. Prove that .

反對稱矩陣的伴隨矩陣 (adjugate) 是對稱或反對稱矩陣。 Let be an skew-symmetric matrix. Prove that is a symmetric matrix for odd and a skew-symmetric matrix for even .

計算 的導數。 Let be an matrix, where each entry is a differentiable function of . Prove that , where is identical to except that the entries in the column are replaced by their derivatives, i.e., if , if .

這是關於分塊矩陣行列式的計算問題。 Let , where and are square matrices of order and , respectively. Let be an matrix and be an matrix. Prove the following identities. (a) . (b) .

這是利用行列式證明一特殊矩陣型態必定可逆的問題。 For any matrix , show that there exists a matrix such that is nonsingular.

如果 和 的所有元為整數，則 有甚麼性質？ Let be an matrix. If all entries of and are integers, show that .