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  =  (a_ij)_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  =  [a_ij]_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  =  [a_ij]_mxm

is a square matrix of order m.

The elements a₁₁, a₂₂, a₃₃ ....... a_mm  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  =  [a_ij] _mxm 

said to be a diagonal matrix if a_ij  =  0 for all i ≠ 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  =  [a_ij] _mxm

is said to be a scalar matrix if

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 I_n. For example,

are unit matrices of orders 2 and 3 respectively.

In general, a square matrix A = [a_ij] _nxn is a unit matrix if

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 A^T (read as A transpose).

For example,

In general, if A  =  [a_ij] _mxn then A^T  =  [b_ij]_nxm where

b_ij  =  a_ij

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

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us : 

v4formath@gmail.com

We always appreciate your feedback. 

You can also visit the following web pages on different stuff in math. 

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

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