Others

Probability / Statistics

Probability Permutation / Combinations Permutations Combinations Binomial Theorem Sample Space Independent Con […]

Calculus

Precalculus Square Root Absolute Values Quadratic Functions Complex Numbers Complex Plane De Moivre’s Th […]

Proof | Property of Transposed Matrix (5)

Proof We use the property of the transposed matrix (4) : $$(AB)^{t}=B^{t}A^{t}$$ Then we have \begin{eqnarray* […]

Proof | Property of Inverse Matrix (2)

Proof Suppose that matrices and are invertible. Then we have \begin{eqnarray*}(AB)(B^{-1}A^{-1})&=&A(B […]

Proof | Property of Inverse Matrix (1)

Proof Suppose that a matrix is invertible. Then there exists such that $$AA^{-1}=A^{-1}A=I$$ where is an ident […]

Cramer’s Rule

Example Use Cramer’s rule to solve the system \begin{eqnarray*}&&3x_{1}-3x_{2}+x_{3}=1\\&&am […]

Proof | Property of Determinant (10)

If A is a triangular matrix, det is the product of the entries on the main diagonal of A: $$\left|\begin{array […]

Proof | Property of Determinant (9)

If a matrix A is inversible, $$|A^{-1}|=\frac{1}{|A|}$$ Proof Let be a identity matrix: $$E=\begin{bmatrix}1 & […]

Proof | Property of Determinant (6)

Proof Suppose that A is an matrix as follows: $$A=\begin{bmatrix}a_{11} &a_{12}&\cdots&a_{1n}\\a_{ […]