# TYPES OF MATRICES

Row Matrix :

A matrix is said to be a row matrix, if it has only one row. A row matrix is also called as a row vector.

For example,

A  =  (5   3   4   1) and B  =  (–3   0   5 )

are row matrices of orders 1 x 4 and 1 x 3 respectively.

In general,

A  =  (aij)1xn

is a row matrix of order 1xn.

Column Matrix :

A matrix is said to be a column matrix, if it has only one column. It is also called as a column vector.

For example,

are column matrices of orders 2x1 and 3x1 respectively.

In general,

A  =  [aij]mx1

is a row matrix of order mx1.

Square Matrix :

A matrix in which the number of rows and the number of columns are equal is said to be a square matrix.

For example,

are square matrices of orders 2 and 3 respectively.

In general,

A  =  [aij]mxm

is a square matrix of order m.

The elements a11, a22, a33 ....... amm  are called principal or leading diagonal elements of the square matrix A.

Diagonal Matrix :

A square matrix in which all the elements above and below the leading diagonal are equal to zero, is called a diagonal matrix.

For example,

are diagonal matrices of orders 2 and 3 respectively.

In general,

A  =  [aijmxm

said to be a diagonal matrix if aij  =  0 for all j.

Note :

Some of the leading diagonal elements of a diagonal matrix may be zero.

Scalar Matrix :

A diagonal matrix in which all the elements along the leading diagonal are equal to a non-zero constant is called a scalar matrix.

For example,

are scalar matrices of orders 2 and 3 respectively.

In general,

A  =  [aij] mxm

is said to be a scalar matrix if

aij = 0, when i ≠ j

aij = k. when i = j

where k is constant.

Unit Matrix :

A diagonal matrix in which all the leading diagonal entries are 1 is called a unit matrix. A unit matrix of order n is denoted by In. For example,

are unit matrices of orders 2 and 3 respectively.

In general, a square matrix A = [aij] nxn is a unit matrix if

aij = 1 if i = j

aij = 0 if ≠ j

A unit matrix is also called an identity matrix with respect to multiplication. Every unit matrix is clearly a scalar matrix. However a scalar matrix need not be a unit  matrix. A unit matrix plays the role of the number 1 in numbers.

Null Matrix or Zero-Matrix :

A matrix is said to be a null matrix or zero-matrix if each of its elements is zero. It is denoted by O.

For example,

are null matrices of order 2x3 and 2x2.

(i) A zero-matrix need not be a square matrix.

(ii) Zero-matrix plays the role of the number zero in numbers.

(iii) A matrix does not change if the zero-matrix of same order is added to it or subtracted from it.

Transpose of a Matrix :

Definition The transpose of a matrix A is obtained by interchanging rows and columns of the matrix A and it is denoted by AT (read as A transpose).

For example,

In general, if A  =  [aij] mxn then AT  =  [bij]nxm where

bij  =  aij

for i  =  1, 2,......n and j  =  1, 2,...... m.

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OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

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Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

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Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

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