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Matrices are the plural form of the word “matrix,” which refers to a rectangular array or table with rows and columns that are used to organise numerical data or other elements. It is possible for them to have any number of columns or rows. Matrices are capable of undergoing a wide variety of operations, including addition, scalar multiplication, multiplication, transposition, and others.
When carrying out these matrices operations, there are a few guidelines that must be adhered to, such as the fact that the matrices can only be added or subtracted if they share the same number of rows and columns, and that they can only be multiplied if the columns in the first matrix and the rows in the second matrix are identical. Let’s get a better understanding of the various matrices and these rules by looking at them in more detail.
Matrices
The arrangement of numbers, variables, symbols, or expressions in a rectangular table that has varying numbers of rows and columns is called a matrix, the plural form of the word matrix. Matrices are also the plural form of the word matrix. They have the shape of rectangles and can be used for a variety of operations, including addition, multiplication, and transposition. The components of the matrix, whether they be numbers or entries, are referred to as its elements. In a matrix, the horizontal entries are referred to as rows, and the vertical entries are referred to as columns.
The Definition of the Matrix
A matrix is a rectangular array of expressions, numbers, symbols, and variables that are defined for mathematical operations such as addition, subtraction, and multiplication. The number of rows and columns in a matrix is what determines its overall size, which is also referred to as the order of the matrix. The order of a matrix that has six rows and four columns is denoted by the notation 6×4, but it is read as 6 by 4. For instance, the provided matrix B is a 3 x 4 matrix and its notation looks like this: [ B ]3×4:
B= [ 2 -5 6 1
8 4 2 -5
4 3 -9 0]
Different Categories of Matrix
The number of elements and the way those elements are arranged in a matrix both play a role in determining the type of matrix that it is.
•A row matrix is a matrix that only has one row in it. This type of matrix is referred to as a row matrix. Example: [1, −2, 4].
•A matrix with a single column is referred to as a column matrix. The term “column matrix” refers to this type of matrix. Example: [−1, 2, 5]T.
•The term “square matrix” refers to a matrix that has the same number of rows and columns throughout its entirety. Take, for instance, the expression
B= [ 2 -5 6
8 4 2
4 3 – 9]
•The term “rectangular matrix” refers to a matrix that does not have an equal number of rows and columns. Take, for instance:
B= [ 2 -5 6
8 4 2]
•The term “diagonal matrix” refers to a matrix in which all of the elements that are not diagonal are replaced by the value zero.
Example:
B= [ 2 0 0
0 4 0
0 0 – 9]
•Identical matrices are referred to as identity matrices. An identity matrix is a diagonal matrix in which each of the diagonal elements is equal to 1.
Example:
B= [ 1 0 0
0 1 0
0 0 1]
•Matrices that are symmetric as well as skew-symmetric:
Matrices that are symmetric: If and only if the matrix DT = D, will we consider the square matrix D to be symmetric. For instance, the matrix
D = [2 3 6
3 4 5
6 5 9]
is considered a symmetric matrix due to the fact that
DT = [2 3 6 =D
3 4 5
6 5 9]
Matrices with skew symmetry A matrix F with the dimensions n by n is said to have skew symmetry if and only if the equation FT = – F holds true.
The reason why the matrix
F = [ 0 3
-3 0]
is considered to be skew-symmetric is as follows:
FT = [ 0 -3
3 0]
-F = [ 0 -3
3 0]
•Any square matrix, A, is referred to as an invertible matrix if there is another matrix, B, such that AB = BA = In, where In is an identity matrix that has n rows and n columns.
•A square matrix A is said to be orthogonal if its transpose equals its inverse. An orthogonal matrix can be any square matrix. i.e., AT = A-1.
Conclusion
The arrangement of numbers, variables, symbols, or expressions in a rectangular table that has varying numbers of rows and columns is called a matrix, the plural form of the word matrix. Matrices are also the plural form of the word matrix. They have the shape of rectangles and can be used for a variety of operations, including addition, multiplication, and transposition. The components of the matrix, whether they be numbers or entries, are referred to as its elements. A matrix is a rectangular array of expressions, numbers, symbols, and variables that are defined for mathematical operations such as addition, subtraction, and multiplication. The number of rows and columns in a matrix is what determines its overall size, which is also referred to as the order of the matrix.
