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 .
Let be vectors in () such that for . Prove that any of these vectors form a basis of .
證明嚴格對角佔優 (strictly diagonally dominant) 矩陣是可逆矩陣。
Let be an matrix. Prove that if for , then is invertible.
Let be an matrix and be an matrix. Prove that
Note that and denote the column space and nullspace of , respectively.
Find all matrices commuting with , where is the matrix all elements of which are equal to .
Prove the following statements.
(a) Let , where are distinct. If , then is a diagonal matrix.
(b) Let , where are distinct and nonzero, and be an matrix, where . If and , then .