Matriks dan Ruang Vektor : Dasar-dasar Matriks dan Operasi Matriks
Matrix
Definition of a matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix
Example :
Size of matrix: number of rows × number of columns. Example : for the matrices above: 2×2, 1×3, 3×3, 1×1, respectively
Matrix with only one row is called row matrix. Thus, matrix with only one column is called column matrix. We will use capital letters to denote matrices and lowercase letter to denote numerical quantities
Example :
Entry that occurs in row i and column j of matrix A will be denoted by a_ij
Square Matrix
A matrix A with n rows and n columns is called a square matrix of order n. The shaded entries a_11,a_22,…,a_nn are said to be on the main diagonal of A
Equality of Matrices
Two matrices are defined to be equal if they have same size and their corresponding entries are equal
Example :
Exercise 1
Let
What is x and y so that A = B ?
Addition and Subtraction
If A and B are matrices of the same size :
Consider the matrices
Then
The expression A+C, B+C, A-C, and B-C are undefined
Scalar Multiples
Syntax for Scalar Multiples :
Syntax for Scalar Multiples :
Example : For the matrices
We have
Multiplying Matrices
If A is an m×r matrix and B is an r×n matrix, then the product AB is the m×n matrix whose entries are determined as follows. To find the entry in row i and column j of AB, we can do :
- Single out row i from the matrix A and column j from the matrix B
- Multiply the corresponding entries from the row and column together
- Add up the resulting products
The product of two matrices is defined if the inside numbers are the same
Example 1
Consider the matrices
For example, the entry in row 2 and column 3 of AB :
Example 2
Consider a system of m linear equations in n unknowns :
The equations above can be written as a
The matrix A is called the coefficient matrix
Transpose Matrix
If A is any m×n matrix, then the transpose of A (A^T ): the n×m matrix that results by interchanging the rows and columns of A;
Example :
Exercise 2
Determine A^T!
Trace
If A is a square matrix, then the trace of A (tr(A)): sum of the entries on the main diagonal of A.
Example :
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