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 $=5 \times 4 \times 3 \times 2 \times 1=120$
$3 \sqrt{3}$
