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Published byMartina O’Connor’ Modified over 4 years ago

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Matrix Definition A Matrix is an ordered set of numbers, variables or parameters. An example of a matrix can be represented by: The matrix is an ordered set of numbers arranged in rows and columns. The number of rows and columns of a matrix defines the matrix dimension. In our example, the matrix has dimension (2x3) -> 2 rows by 3 columns. WARNING: to assign a dimension to the matrix always starts indicating the number of rows, and then the number of columns. The numbers of the matrix are enclosed in brackets.

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Matrix Content Each number in the matrix is characterized by a position expressed by the row and column to which it refers. For instance, the number 4 corresponds to the element in the matrix A located at the intersection of the second row and second column. The components of the matrix (numbers, parameters, variables) are the matrix elements positioned at the intersection of each row and each column. To each matrix is associated a capital letter in bold type that expresses the matrix name. The letter A assumes the content of the matrix. Recalling the letter in the calculations of matrix , the same ordered set of numbers in square brackets is implicitly recalled.

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Matrix Dimension The same matrix can be also written in the following way where the dimension is indicated in parenthesis below the capital letter. The matrix A of dimension 2 by 3 is a rectangular matrix. A rectangular matrix has a number of rows different from the number of columns. While a matrix B (2 by 2) is a square matrix. A square matrix has a number of rows equal to the number of columns. An example of a square matrix is as follows:

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Matrix Elements A matrix of dimension n by m (n rows by m columns ) can be represented in the following general way: Within the matrix we can find the elements, each one characterized by a row index and a column index which define its position. The element a12 identifies that element on the intersection of the first row and second column.

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Vectors If a matrix contains a single column takes the name of column vector, while if it contains a single row is called row vector. A vector is a matrix whose dimension of row or column is equal to 1. For example, the matrix is a column vector. Vectors are usually denoted by a lowercase letter in bold, while the matrix by a capital letter always bold.

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Element Order Two matrices are equal if and only if the values and positions of the elements in the two matrices are equal. The matrices A and B are equal, while the matrix C even if it contains the same values is not equal to the other two because the order of elements diverges. The order of elements in a matrix is very important!!

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Addition of Matrices Two matrices can be added if and only if they have the same dimension. If the two matrices have the same dimension, the matrices are said to be conformable for addition. The sum of two matrices is performed by adding the elements of the two matrices that are in the same position. The same rules outlined above apply to the subtraction:

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**Multipication of Matrix by a Scalar**

A matrix can be multiplied by a scalar, ie for a constant. The result is a matrix whose elements will be obtained through the product of the elements of the initial matrix by the scalar: Scalar

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**Multiplication of Matrices**

The product of two matrices can be done only if the two matrices are conformable for multiplication. Two matrices are conformable for multiplication when the number of columns in the first matrix is equal to the number of rows in the second matrix. In other words, the column dimension of the first matrix must be equal to the row dimension of the second matrix. Given two matrices: The two matrices are conformable for the multiplication, because the number of columns of A is equal to the number of rows of B. Warning: the two matrices are no longer conformable for the multiplication if B is prepended to A. Again, the order is important!!

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**Multiplication of Matrices**

The result of the product between two matrices conformable for multiplication is a matrix with a row dimension equal to the number of row of the first matrix and a column dimension equal to number of columns of the second matrix. Found that two matrices are conformable for multiplication, it is possible to calculate the product following the rule "row by column". Each product “row by column” identifies the element in the resulting matrix positioned on the row and column used for the calculation. Then, the product “row by column" concerning the first row element of A and first column element of B identifies the element in the first row and first column of the resulting matrix.

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**Multiplication of Matrices**

The operation “row-by-column" is the sum of the products of the elements of row of the first matrix by the corresponding elements of column of the second matrix.

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Square Matrix The square matrix is a matrix that has the same number of rows and columns. are two square matrices.

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Diagonal Matrix A matrix is said diagonal when it presents non-zero values only on the main diagonal: The principal diagonal in a matrix identifies those elements of the matrix running from north-west to south-east. The secondary diagonal of a matrix identifies those elements of the matrix running from north-east to south-west. Principal diagonal Secondary diagonal

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Identity Matrix The identity matrix is a matrix that presents unit values on the principal diagonal and zero values elsewhere . The product between a matrix A and an identity matrix I produces the same matrix A.

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**Inverse Matrix An inverse matrix can be written as:**

Above the inverse matrix of A. A matrix multiplied by its inverse produces as a result the identity matrix:

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Transposed matrix A matrix is said transposed when its rows become columns and its columns become rows. Given a matrix: Its transposed matrix is: To indicate that a matrix is transposed is used to put an apostrophe at the upper right hand side of the letter indicating the matrix.

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**Some operations between vectors and matrices**

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**Some operations between vectors and matrices**

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