MATRIX ALGEBRA

In this section we will discuss the basic operations of matrices. We start with the idea of matrix equality before taking up the operations of addition and multiplication. In matrix algebra, the elements are ordered numbers and therefore operations on them have to be done in ordered manner. It may be useful to note that while we deal with the main operations such as addition and multiplication. Other operations viz., subtraction and division are derived out of those.

 



Equality of Matrices

 

Two matrices are equal if the following three conditions are met:

i) Each matrix has the same number of rows.

ii) Each matrix has the same number of columns.

iii) Corresponding elements within each matrix are equal.

The above conditions simply require that matrices under consideration are exactly the same.

 

Important Key Notes :-

 

Diagonal Matrix: Non-zero elements only in the diagonal running from the upper left to the lower right.

 

Equality of Matrices: Two matrices are equal if each matrix has the same number of rows, columns and corresponding elements within each are also equal.

 

Identity matrix: A matrix usually written as I, with 1 (ones) on the main diagonal and zeros elsewhere.

 

Lower Triangular Matrix: A special kind of square matrix with all its entries above the main diagonal aszero.

 

Matrix Multiplication: A feasible operation when the number of columns in a first matrix is equal to number of rows in a second matrix.

 

Matrix: A way of representing data in a rectangular array.

 

Negation of a Matrix: Elements of a matrix with their replacement by their negation.

 

Orthogonal Matrix: Matrix A is called orthogonal matrix if AA’ = A’A = I

 

Rectangular Matrix: A matrix with the number of rows not equal to the number of columns. 

 

Scalar Matrix: A diagonal matrix in which all the diagonal elements are the same.

 

Scalar: A single constant, variable, or expression.

 

Skew Symmetric Matrix: Matrix A is called Skew Symmetric matrix if A’ = -A

 

Square Matrix: A matrix in which the number of rows is equal to the number of columns.

 

Sub Matrix: A matrix obtained by deleting some rows or columns or both of a given matrix is called sub matrix of a given matrix.

 

Symmetric Matrix: A matrix is symmetric if it equals its own transpose. 

 

Dimension(s) or Order:  The number of rows and the number of columns in a matrix. 

 

Transpose of Matrix: New matrix obtained by interchanging the rows and columns of the original.

 

Upper Triangular Matrix: A special kind of square matrix with all its entries below  the main diagonal as zero.

 

Zero (or null) Matrix: Matrix whose elements are all zeros.

 

Cofactor: The signed minor.

 

Cramer’s Rule: Methodof solving a system of n linear equations in n variables using determinants.

 

Determinant: A numerical value computed from the elements of a square matrix

 

Linear Equation System: A collection of two or more linear equations involving the same number of variables.

 

Minor: Value obtained from the determinant of a square matrixby deleting out a row and a column corresponding to the element of a matrix.

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