Pengantar Logika : Proposisi, Hubungan Antar Proposisi, Definisi Term, Sifat Term, dan Contoh Soal

Matriks dan Ruang Vektor : Inversi Matriks, Properti dalam Matriks, dan Bentuk Matriks

Properties of Matrix Arithmetic

Assuming that the sizes of the matrices are such that the indicated operations can be performed, the following rules of matrix arithmetic are valid

1. A+B = B+A ( commutative law for addition )
2. A+(B+C) = (A+B)+C ( Associative law for addition )
3. A(BC) = (AB)C ( Associative law for multiplication )
4. A(B+C) = AB+AC ( left distributive law )
5. (A+B)C = AC+BC ( right distributive law )
6. a(B+C) = aB+aC
7. (a+b)C = aC+bC
8. a(bC) = (ab)C
9. a(BC) = (aB)C = B(aC)

Identity Matrix

A square matrix with 1’s on the main diagonal and zeros elsewhere is called an identity matrix

 

Example

AI = IA = A 

Inverse Matrix

A is a square matrix, and if a matrix B of the same size can be found such that AB=BA=I, then 

• A is invertible (nonsingular) 
• B is an inverse of A (A is an inverse of B) 
• If C is also an inverse of A, then B=C 

If no such matrix B can be found → A is singular. If A is invertible, the its inverse will be denoted by A^(-1)

The matrix of

is invertible if and only if ad-bc ≠ 0 and

Example 1

Determine whether the matrix is invertible. If so, find its inverse.

Some Theorem of Inverse Matrices

If A and B are invertible matrices with the same size, then AB is invertible and (AB)^(-1)=B^(-1) A^(-1)

If A is invertible and n is a nonnegative integer, then:

• A^(-1) is invertible and (A^(-1) )^(-1)=A
• A^n is invertible and (A^n )^(-1)=A^(-n)=(A^(-1) )^n
• kA is invertible for any nonzero scalar k, and (kA)^(-1)=k^(-1) A^(-1)

If the sizes of the matrices are such that the stated operations can be performed, then:

• ( A^T  )^T = A
• ( A+B )^T = A^T + B^T
• ( A-B )^T = A^T- B^T
• ( kA )^T = kA^T
• ( AB )^T = B^T A^T

Elementary Matrices and a Method for Finding A^(-1)

Some Equivalent Statements

If A is an m×n matrix, then the following statements are equivalent, that is, all true or all false

1. A is invertible
2. Ax=0 has only the trivial solution
3. The reduced row echelon form of A is I_n

Inversion Algorithm

To find the inverse of an invertible matrix A : 

1. Find a sequence of elementary row operations that reduces A to the identity
2. Perform that same sequence of operations on I_n to obtain A^(-1)

Example

Find the inverse of

Solution

Exercise 1

Determine A^(-1) (if exist) ! 

Inverse and Solution of Linear System

If A is an invertible n×n matrix, then for each n×1 matrix b, the system of equations Ax=b has exactly one solution, namely 

x=A^(-1) b

Exercise 2

Solve the linear equations using A^(-1)

Diagonal, Triangular, and Symmetric Matrices

Diagonal Matrices

A square matrix in which all the entries off the main diagonal are zero is called a diagonal matrix. 

Example

A general n×n diagonal matrix D can be written as

Example 2

If

Determine the value of A^(-1), A^5,A^(-5) !

Solution

Upper and Lower Matrices

A square matrix A=[a_ij] is upper triangular if and only if all entries to the left of the main diagonal are zero; that is, a_ij=0, if i > j

Symmetric Matrices

A square matrix A is said to be symmetric if A = A^T 

(A)_ij = (A)_ji 

Example

Sumber

http://informatika.unpar.ac.id/

Slide MRV : Properties of Matrices and Inverse