Matrices I

  • Introduction

A matrix (plural : matrices) is a rectangular arrangement of m \times n numbers in m rows and n columns. It is written in between a pair of rectangular brackets. The numbers belonging to a matrix are known as its elements. Matrices are represented by capital Roman letters. For example, consider the following matrix M.

{M = \begin{bmatrix} 2& 3 & 0 \\[0.3em] 1 & -3 & \sqrt 2 \end{bmatrix}}

It has 2 rows and 3 columns. Hence it is a 2 \times 3 matrix.

If a matrix has m rows and n columns, it is said to be of the order m by n. So, M is of the order 2 by 3 and has 6 elements.

  • Representation of an Element

a_{ij} corresponds to that element of a matrix, which is present in the ith row and jth column of it. So, for matrix M,

{a_{21} = 1, a_{13}= 0, a_{23}= \sqrt 2}

  • Transpose of a matrix

A matrix obtained by interchanging the rows and columns of a matrix is known as the transpose of the matrix. So,

M^T = \begin{bmatrix} 2& 1 \\[0.3em] 3 & -3 \\[0.3em] 0 & \sqrt 2 \end{bmatrix}

  • Equality of 2 matrices

2 matrices A and B are equal iff (if and only if)

i) their order is same and

ii) \forall a in A and b in B, a_{ij}= b_{ij}

  • Row Matrix

A row matrix has only 1 row and more than 1 column. e.g.

{P = [1 \ 3 \ 2]}

  • Column Matrix

A column matrix has only 1 column and more than 1 row. e.g.

{Q = \begin {bmatrix} 2 \\ 3 \\ 5 \\ 6 \end {bmatrix}}

  • Null Matrix

A null matrix is the one, whose all elements are 0. e.g.

{R = \begin {bmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end {bmatrix}}

  • Square Matrix

A square matrix has equal number of rows and columns. So, the total number of elements in a square matrix is always a perfect square. e.g.

{S = \begin {bmatrix} 1 & 4 & \sqrt 5 & -6 \\ 77 & 64 & \frac 1 2 & 0 \\ e & \pi & 5 & 3 \\ 0 & 1 & 1 & 4 \end {bmatrix}}

So, S has 4 rows as well as 4 columns, so 16 elements.

The elements a_{ij} of a square matrix, for which i=j are known as the diagonal elements. Thus, 1,64, 5 and 4 are diagonal elements of S.

  • Diagonal Matrix

A square matrix, whose all elements are zero, except the diagonal elements, is known as the diagonal matrix.e.g.

{D = \begin {bmatrix} 3 & 0 & 0 \\ 0 & -8 & 0 \\ 0 & 0 & 13 \end {bmatrix}}

  • Identity Matrix

An identity matrix is a square matrix, whose diagonal elements are equal to 1 and all other elements are 0. This matrix is of special importance in the matrix theory. Similar to the number 1 in the theory of numbers, it acts as a unit entity. It is generally denoted by I. So, for any square matrix A, AI = A. Note that A and I must have the same order.

{I_{2} = \begin {bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}}

{I_{3} = \begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {bmatrix}}

  • Upper- and Lower-Triangular Matrices

An upper-triangular matrix is the one, whose elements below the diagonal are 0. Similarly, a lower-triangular matrix is the one, whose elements above the diagonal are 0.

{U = \begin {bmatrix} 1 & 5 & 6 \\ 0 & 3 & 9 \\ 0 & 0 & 34 \end {bmatrix}}

{L = \begin {bmatrix} 6 & 0 & 0 \\ 2 & 2 & 0 \\ 3 & 5 & 5 \end {bmatrix}}

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