1708Yes Aditiya
But actually Question is How many 3 *3 matrix which satisfy the given Condition.
\hspace{-16}$Find total no. of matrices for which Inverse of a matrix is exists\\\\ If its element are taken from the set $\bf{\left\{0,1\right\}}$
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6 Answers
262Identity matrix of any order will satisfy the condition
so there should be infinitely many such matrices.
1708
49The inverted matrix of any matrix A is given by:
A-1=1|A|.adj(A)
the matrix becomes non-invertible if the matrix A-1 is undefined..
Or in other words if |A|=0
thus the solutions for which the determinant is not zero gives the answer.
The determinant can be zero if the entire column or row has zero as the element.
The cases are:
1) Only 1 row is fully zero. [3x64 - 3x(8-1) - 1 ways]
2) Two rows are fully zero [3x(8-1) ways]
3) Everything is zero. [1 way]
Can you think of any other way in which the |A| is zero but no row or column is and finish the problem?
1yes if 3 rows will be fully 1,then also 0.
111
111
111
1(1-1)-1(1-1)+1(1-1)=0
if there will be
100
011
011
here the left two zeroes can be replaced by 01,10,11.(still |A| would be zero)
so total 4 cases.for each case there will be 3 cases.so total =12
it will be zero too.
110
111
111
here i think there will be 9 cases for the 9 positions of 0.
12+9+1=22
now if we take 3 zeroes we get the cases which subhomoy bhaiya has considered.
so answer=29-[3x64 - 3x(8-1) - 1+3x(8-1)+1+22 ]
i guess i am right on this one :P