Prove that the value of the determinant
` |{:(-7,,5+3i,,(2)/(3)-4i),(5-3i ,,8,,4+5i),((2)/(3) +4i,,4-5i,,9):}|" is real "`

Explanation

Let
` z=|{:(-7,,5+3i,,(2)/(3)-4i),(5-3i ,,8,,4+5i),((2)/(3) +4i,,4-5i,,9):}|`
To Prove that this number (z) is real we have to prove that `hatz=z`. Now we know that conjugate of complex number is distributive over all algebraic operations . Hence to take conjugate of z in (1) we need not to expand determinant.
to get the conjugate of z we can take conjugate of each element of determinant . Thus
` hatz=|{:(-7,,5+3i,,(2)/(3)-4i),(5-3i ,,8,,4+5i),((2)/(3) +4i,,4-5i,,9):}|" "(2)`
Now interchanging rows into columns (taking transpose) in (2)
` " we have "hatz=|{:(-7,,5+3i,,(2)/(3)-4i),(5-3i ,,8,,4+5i),((2)/(3) +4i,,4-5i,,9):}|" "(3)`
`" or " harz =z" " ["from (1) and (3)"] (4)`
Hence z is purely real.