线性代数chapter过程稿.ppt

§3.2 Properties of Determinants If there are r interchanges, then Theorem 3 shows that Thus §3.2 Properties of Determinants Theorem 4 A square matrix A is invertible if and only if det A ≠ 0. Example : Compute det A, where Solution: Compute §3.2 Properties of Determinants Theorem 5 If A is an n×n matrix, then det AT = det A. Each statement in Theorem 3 is true when the word “row” is replaced by “column”. Proof: when n = 1, the theorem is obvious. suppose the theorem is true for k*k eterminants and let n = k+1 the cofactor of a1j in A equals the cofactor of aj1 in AT. Hence the cofactor expansion of det A along the first row equals the confactor expansion of det AT down the first column. That is A and AT have equal determinants. By the principle of induction, the theorem is true for all n≥1. §3.2 Properties of Determinants 3. Determinants and Matrix Products Theorem 6 If A and B are n×n matrices, then det AB = (det A)(det B). If A is invertible, then A and the identity matrix In are row equivalent by the Invertible Matrix Theorem. So there exist elementary matrices such that For brevity, write for det A. Then repeated application of Theorem 3, as rephrased above, show that §3.2 Properties of Determinants Example ：Verify Theorem 6 for Solution: ∵ ∴ warning: det(A + B) != detA + detB Example： Example ： =125 Sol. 4. A Linearity Property of the Determinant Function Suppose that the jth columns of A is allowed to vary, and write Define a transformation T from Rn to R by Then, T(cx) = cT(x), for all scalars c and all x in Rn T(u + v) = T(u) + T(v) , for all u, v in Rn §3.3 Cramer’s Rule, Volume, and Linear Transformations 1. Cramer’s Rule 2. A Formula for A-1 3. Determinants as Area or Volume 4. Linear Transformations §3.3 Cra