Kategóriák: Minden - matrices - equality - addition

a David Kedrowski 14 éve

188

MAT.116 5.4

Matrices are fundamental elements in linear algebra and are utilized in various mathematical and real-world applications. They are ordered rectangular arrays of numbers organized into rows and columns.

MAT.116 5.4

MAT.116 5.4 Matrices

Addition and Subtraction

If A and B are two matrices of the same size, then:

  • The sum A+B is the matrix obtained by adding the corresponding entries in the two matrices.
  • the difference A-B is the matrix obtained by subtracting the corresponding entries in B from those in A.
  • Zero Matrix

    A zero matrix is one in which all entries are zero.

    A zero matrix O has the property that A+O = O+A = A for any matrix A having the same size as that of O.

    Laws for Matrix Addition

    If A, B, and C are matrices of the same size, then

  • A+B = B+A (commutative law)
  • (A+B)+C = A+(B+C) (associative law)
  • Definitions

    Transpose of a Matrix

    If A is an m x n matrix with elements a_{ij}, then the transpose of A is the n x m matrix A^T with elements a_{ji}.

    Scalar Multiplication

    If A is a matrix and c is a real number, then the scalar product cA is the matrix obtained by multiplying each entry of A by c.

    Equality of Matrices

    Two matrices are equal if they have the same size and their corresponding entries are equal.

    Using Matrices to Represent Data

    Special Matrices
    Square matrix
    Column matrix
    Row matrix
    Definition
    Block / Table / Rectangular Array

    A matrix is an ordered rectangular array of numbers. A matrix with m rows and n columns has size m x n. The entry in the ith row and the jth column of a matrix A is denoted by a_{ij}.

    Size = rows x columns
    Columns
    Rows
    Entries