Algebra of Matrices

A matrix (plural: matrices) is an arrangement of numbers, expressions or symbols in a rectangular array. This arrangement is done in horizontal-rows and vertical-columns, having an order of number of rows x number of columns.

Algebra of Matrices is the branch of mathematics, which deals with the vector spaces between different dimensions. The innovation of Algebra of Matrices came into existence because of n- dimensional planes present in our coordinate space.

Every pair of points in a Three-dimensional space represent a unique equation with one or more than one solution.

The basic idea or the central idea of applied mathematics revolve around Linear Algebra. For instance, by reliving the rules and regulations or rather say Axioms, we get into generalization of vector space, which by calculus leads to the Solution of Differential Equations.

Algebra of Matrix

Algebra of matrix involves operation of matrices, such as Addition, subtraction, multiplication etc.

Let us understand the operation of matrix in a much better way-

Addition/Subtraction of matrices

Two matrices can be added/subtracted, iff (if and only if) the number of rows and columns of both the matrices are same, or the order of the matrices are equal.

For addition/subtraction, each element of the first matrix is added/subtracted to the element present in the 2nd matrix.

Example: \(\begin{bmatrix} 2 & 0 & 5\\ 3 & 2 & 9 \end{bmatrix} + \begin{bmatrix} 7 & 4 & 1 \\ 8 & 13 & 0 \end{bmatrix} = \begin{bmatrix} 9 & 4 & 6 \\ 11 & 15 & 9 \end{bmatrix}\)

Matrix Multiplication

Like Matrix can be Multiplied two ways,

(i) Scalar Muliplication

(ii) Multiplication with another matrix:

Scalar Multiplication – It involves multiplying a scalar quantity to the matrix. Every element inside the matrix is to be multiplied by the scalar quantity to form a new matrix.

For example-

\(5 \times \begin{bmatrix} 5 & 7\\ 12 & 3 \\ 6 & 2 \end{bmatrix} = \begin{bmatrix} 25 & 35\\ 60 & 15 \\ 30 & 10 \end{bmatrix}\)

Multiplication of a matrix with another matrix – Two matrix can be multiplied iff the number of column of the first matrix is equal to the number of rows of the second matrix.

Consider two matrix M1 & M2, having order of \( m_{1} \times n_{1},\) and \(m_{2} \times n_{2},\).

The matrices can be multiplied if and only if \(n_{1} = m_{1}\).

The matrices, given above satisfies the condition for matrix multiplication, hence it is possible to multiply those matrix.

The resultant matrix obtained by multiplication of two matrices, is the order of \(m_{1}, n_{2}\), where \(m_{1}\) is the number of rows in the 1st matrix and \(n_{2}\) is the number of column of the 2nd matrix.