1. Row Matrix 2. Column Matrix 3. Square Matrix 4. Diagonal Matrix 5. Scalar Matrix 6. Identity (or) Unit Matrix 7. Zero matrix (or) null matrix 8. Transpose of a matrix 9. Triangular Matrix

**Types of Matrices**

In this section, we shall define certain types of matrices.

A matrix is said to be a row matrix if it has only one row and any number of
columns.

A row matrix is also called as a row vector.

For example, *A* = (8 9 4 3), B = (-√3/2 1 √3) are row
matrices of order 1×4 and 1×3 respectively.

In general A = (*a*_{11}
*a*_{12} *a*_{13} … *a*_{1n}) is a row matrix of order
1×*n*.

A matrix is said to be a column matrix if it has only one column and any number of
rows. It is also called as a column vector.

For example, are column matrices of order 3 ×1, 2 ×1 and 4 ×1 respectively.

In general, is a column matrix of order m × 1.

A matrix in which the number of rows is equal to the number of columns is called a square matrix. Thus a matrix *A* = (*a _{ij}* )

For example, are square matrices.

In general, are square matrices of orders 2 × 2 and 3 × 3 respectively.

*A *=* *(*a _{ij} *)

**Definition : **In a square matrix,
the elements of the form** ***a*_{11},** ***a*_{22},** ***a*_{33}, . . . (i.e)** ***a _{ii}*

A square matrix, all of whose elements, except those in the
leading diagonal are zero is called a diagonal matrix.

(ie) A square matrix *A* = (*a _{ij}* ) is said
to be diagonal matrix if

For example, are diagonal matrices.

A diagonal matrix in which all the leading diagonal elements are
equal is called a scalar matrix.

In general, *A *=* *(*a _{ij} *)

A square matrix in which elements in the leading diagonal are all
“1” and rest are all zero is called an identity matrix (or) unit matrix.

Thus, the square matrix *A *= (*a*_{ij}) is an identity matrix if *a _{ij} *=

A unit matrix of order *n* is written as *I _{n.}*

** **are identity matrices of order 2 and 3
respectively.

A matrix is said to be a zero matrix or null matrix if all its elements are zero.

For example, are all zero matrices of order 1×1,
2 × 2 and 3 × 3 but of different orders.
We denote zero matrix of order *n* ×*n *by O_{n.}

is a zero matrix of the order 2×3.

The matrix which is obtained by interchanging the elements in rows
and columns of the given matrix *A* is called transpose of *A* and is denoted
by *A ^{T}* (read as

If order of *A* is *m* ×*n* then order of *A ^{T}*
is

We note that (*A ^{T}* )

A square matrix in which all the entries above the leading
diagonal are zero is called a lower triangular matrix.

If all the entries below the leading diagonal are zero, then it is
called an upper
triangular matrix.

**Definition: **

A square matrix** ***A*** **=** **(*a _{ij}*

For example, A = is an upper triangular matrix and B = is a lower triangular matrix.

Two matrices *A* and *B* are said to be equal if and
only if they have the same order and each element of matrix *A* is equal
to the corresponding element of
matrix *B*. That is, *a* * _{ij }*=

For example, if

then we note that *A* and *B* have same order and *a _{ij}*
=

**Progress Check**

1. The number of column(s) in a column matrix are _______.

2. The number of row(s) in a row matrix are _______.

3. The non-diagonal elements in any unit matrix are______.

4. Does there exist a square matrix with 32 elements?

The negative of a matrix *A _{m}*

Additive inverse of an element *k* is -*k* . That is,
every element of –*A* is the negative of the corresponding element of *A*.

For
example, if

**Example 3.53**

Consider the following information regarding the number of men and** **women workers in three
factories I, II and III.

Represent the above information in the form of a matrix. What does
the entry in the second row and first column represent?

*Solution*

The information is represented in the form of a** **3

The entry in the second row and first column represent that there
are 47 men workers in factory II

**Example 3.54**

If a matrix has** **16** **elements, what are the possible orders it can have?

*Solution*

We know that a matrix of order* **m*** **×

Such ordered pairs are (1,16), (16,1), (4,4), (8,2), (2,8)

Hence possible orders are 1 × 16, 16 ×1 , 4 ×4 , 2 ×8 , 8 ×2

**Example 3.55**

Construct a** **3** **×3** **matrix whose elements are** ***a*** *** _{ij }*=

*Solution*

The general 3×3 matrix is given by

Find the value of *a, b, c, d*
from the equation

The given matrices are equal. Thus all corresponding elements are
equal.

Therefore,

*a *−* b *=* *1…(1)

2*a* + *c* = 5…(2)

2*a* − *b* = 0…(3)

3*c* + *d* = 2…(4)

(3) Gives

2*a* − *b* = 0

2*a* = *b *…(5)

Put 2*a* = *b* in equation (1), *a* − 2*a* = 1 gives *a* = −1

Put *a *= −1* *in equation* *(5), 2(− 1) = *b *gives
*b* = −2

Put *a *= −1* *in equation* *(2), 2(− 1) +*c* = 5
gives *c* = 7

Put *c *=* *7* *in equation (4), 3(7) + *d*
= 2 gives *d* = −19

Therefore, *a* = −1, *b* = −2, *c* = 7, *d* =
−19

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