A, B and C are three square matrices of order 3 such that A= diag `(x, y, z)`, det `(B)=4` and det `(C)=2`, where `x, y, z in I^(+)`. If det`(adj (adj (ABC)))``=2^(16)xx3^(8)xx7^(4)`, then the number of distinct possible matrices A is ________ .

Answer

9

Explanation

det. (adj. (adj. (ABC)))`=2^(16)xx3^(8)xx7^(4)`
`|A|^(4)|B|^(4)|C|^(4)=2^(16)xx3^(8)xx7^(4)`
`implies (xyz)^(4)xx2^(8)xx2^(4)=2^(16)xx3^(8)xx7^(4)`
`implies (xyz)^(4)=2^(4)xx7^(4)`
`implies xyz=2^(1)xx7^(1)`
Number of possible solutions `=^(3)C_(2) xx .^(3)C_(2)=9`