# Determinant

• Q) Two non zero distinct numbers a and b are used elements to make determinants of the third order. The number of determinant whose value is zero for all a,b is:-

Determinant of a 3x3 matrix is zero when either any two rows or columns are identical or all the three rows or columns are identical. It also happens when all the elements in any row are 0 but this is ruled out here.

Each element in a row has two choices (a or b) and it implies there are 2∗2∗2 =8 ways to fill a row in a determinant. So, there are a total of 8∗8∗8 =512 different determinants.

Number of determinants such that each row is unique can be formed by selecting 3 combinations out of 8 combinations to fill a row and so, the number of such determinants = 8C3∗3p3 =336 So, the total number of 3x3 determinants with at least two identical rows or columns = 512 - 336 = 176
have missed two cases

1.If one of the rows is aaa and another row is bbb these two rows are linearly dependent and the determinant is 0, 36 cases are here
2.b b b
a b a
b a b
this determinant is 0
since the sum of the last two rows is a constant multiple of the first row. 36 are here.

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• @astha-reddy answer given is 32
Options are
1)24
2)32
3)2^9
4)NOT