Question

b) What is the number of all possible
2x 3 matrices with entries
O or 1
?​

Answer 1

$\large\underline{\sf{Solution-}}$

Let us consider a matrix of order 2 × 3 as

$\rm :\longmapsto\:Let \: A \: = \: \: \begin{bmatrix} a_{11} & a_{12}& a_{13}\\ a_{21} & a_{22}& a_{23}\end{bmatrix}$

Now,

$\rm :\longmapsto\:a_{11} \: can \: take \: the \: value \: either \: 0 \: or \: 1.$

$\rm :\longmapsto\: \therefore \: there \: are \: 2 \: ways \: to \: fill \: the \: place \: a_{11}$

↝ Again,

$\rm :\longmapsto\:a_{12} \: can \: take \: the \: value \: either \: 0 \: or \: 1.$

$\rm :\longmapsto\: \therefore \: there \: are \: 2 \: ways \: to \: fill \: the \: place \: a_{12}$

↝ Again,

$\rm :\longmapsto\:a_{13} \: can \: take \: the \: value \: either \: 0 \: or \: 1.$

$\rm :\longmapsto\: \therefore \: there \: are \: 2 \: ways \: to \: fill \: the \: place \: a_{13}$

↝ Again,

$\rm :\longmapsto\:a_{21} \: can \: take \: the \: value \: either \: 0 \: or \: 1.$

$\rm :\longmapsto\: \therefore \: there \: are \: 2 \: ways \: to \: fill \: the \: place \: a_{21}$

↝ Again,

$\rm :\longmapsto\:a_{22} \: can \: take \: the \: value \: either \: 0 \: or \: 1.$

$\rm :\longmapsto\: \therefore \: there \: are \: 2 \: ways \: to \: fill \: the \: place \: a_{22}$

↝ Again,

$\rm :\longmapsto\:a_{23} \: can \: take \: the \: value \: either \: 0 \: or \: 1.$

$\rm :\longmapsto\: \therefore \: there \: are \: 2 \: ways \: to \: fill \: the \: place \: a_{23}$

Therefore,

↝  By Fundamental Principal of Counting,

$\rm :\longmapsto\:Possible \: number \: of \: matrices = 2 \times 2 \times 2 \times 2 \times 2 \times 2$

$\bf\implies \:Possible \: number \: of \: matrices = {2}^{6} = 64$

Additional Information :-

↝ Let us assume a matrix of order m × n,

where

• m represents number of rows

and

• n represents number of columns,

then

↝ Order of matrix provides the following Information,

• 1. Number of elements in matrices = mn

• 2 Number of elements in each row = number of columns.

• 3. Number of elements in each column = number of rows.

Answer 2

$\huge \purple{ \underline{ \boxed{ \red{ \mathbb{ANSWER : }}}}}$

$\red{ \textsf{The number of all possible matrices of order 2x3 with entry 1 or 0 are 64 .}}$

$\huge \purple{ \underline{ \boxed{ \red{ \mathbb{EXPLANATION : }}}}}$

$\large \green{ \underline{ \underline{ \mathbb{GIVEN : }}}}$

$\orange{ \textsf{A matrix with order 2 x 3 :}}$

$\orange{\left[ \begin{array}{c c c} \bf{a11}&\bf{a12}& \bf{a13} \\ \bf{a21}&\bf{a22}&\bf{a23} \end{array}\right]}$

$\orange{ \textsf{Having entries 1 and 0 .}}$

$\large \green{ \underline{ \underline{ \mathbb{SOLUTION : }}}}$

$\red{ \textsf{a11 can have two entries 1 and 0 .}} \\ \red{ \textsf{a12 can have two entries 1 and 0 .}} \\ \red{ \textsf{a13 can have two entries 1 and 0 .}} \\ \red{ \textsf{a21 can have two entries 1 and 0 .}} \\ \red{ \textsf{a22 can have two entries 1 and 0 .}} \\ \red{ \textsf{a23 can have two entries 1 and 0 .}}$

$\red{ \textsf{Therefore , The number of all possible matrices of order 2 x 3 with entry 1 or 0 are :}}$

$\red{ \longrightarrow{ \mathbb{2 \times 2 \times 2 \times 2 \times 2 \times 2}}}$

$\red{ \longrightarrow{ \mathbb{ {2}^{6} }}}$

$\red{ \longrightarrow{ \mathbb{64}}}$

$\large \green{ \underline{ \underline{ \mathbb{KNOW \: MORE : }}}}$

$\orange{\mathbb{MATRIX : }}$

A matrix is an rectangular array having 'm' number of rows and 'n' number of columns .

$\orange{ \mathbb{ORDER \: OF \: MATRIX :}}$

A matrix having 'm' number of rows and 'n' number of columns is said to be matrix of order m x n .