91影视

A matrix A of order n x n is said to be invertible if and only if the determinant of matrix A is zero.

If A is a matrix of order n x n then the total number of the matrices whose entries are all 0鈥檚 and 1鈥檚 is$2^{n^2}.$

Here we have the matrix A of order 3 x 3 then the total number of the matrices whose entries are all 0鈥檚 and 1鈥檚 is$2^9$ .

A matrix A of order 3 x 3 is non-invertible if either all elements of one row are zero or two rows of matrix A are identical. Therefore for an invertible matrix A we have-

1. The total number of choices of elements of the first row whose entries are all 0鈥檚 and 1鈥檚 is$2^3-1$ .
2. The total number of choices of elements of the second row whose entries are all 0鈥檚 and 1鈥檚 is$2^3-2$ .
3. The total number of choices of elements of the third row whose entries are all 0鈥檚 and 1鈥檚 is$2^3-4.$

The total number of the matrices whose entries are all 0鈥檚 and 1鈥檚 is$2^9$ .

The total number of matrices whose entries are all 0鈥檚 and 1鈥檚 and are invertible =

$\left(2^3-1\right)脳\left(2^3-2\right)脳\left(2^3-4\right)=7脳6脳4=168$ .

Probability of all invertible matrices whose entries are all 0鈥檚 and 1鈥檚

$=\frac{168}{2^9}=\frac{168}{512}鈮�0.33$

Thus, among the 3 脳 3 matrices whose entries are all 0鈥檚 and1鈥檚, most are not invertible.

Since the probability of all invertible matrices whose entries are all 0鈥檚 and 1鈥檚 is approximately $0.33$and hence, among the 3 脳 3 matrices whose entries are all 0鈥檚 and1鈥檚, most are not invertible.