### Matrices Question 2

#### Question 2 - 24 January - Shift 2

The number of square matrices of order 5 with entries from the set ${0,1}$, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1 , is

(1) 225

(2) 120

(3) 150

(4) 125

## Show Answer

#### Answer: (2)

#### Solution:

#### Formula: Properties of Adjoint of a Matrix, Properties of Inverse of a matrix

In each row and each column exactly one is to be placed -

$\therefore$ No. of such materials $=5 \times 4 \times 3 \times 2 \times 1=120$

Alternate :

$ \begin{bmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0\end{bmatrix} $

Step-1 : Select any 1 place for 1’s in row 1.

Automatically some column will get filled with 0 ’s.

Step-2 : From next now select 1 place for 1’s.

Automatically some column will get filled with 0 ’s.

$\Rightarrow$ Each time one less place will be available for

putting 1’s.

Repeat step-2 till last row.

Req. ways $=5 \times 4 \times 3 \times 2 \times 1=120$

$3 \sqrt{3}$