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 = [aij] mxm
said to be a diagonal matrix if aij = 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 = [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 i ≠ 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
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 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
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