Question

determinants show that​

Answer 1

Answer:

These two determinants are equal.

Step-by-step explanation:

The determinant of a matrix is found out by selecting a row or column and then find the minor of each element in that row or column and then multiply the corresponding element with the corresponding minor ,finally do alternate addition and subtraction.

Minors are found by eliminating the column and row of the element considered, and find the determinant of the remaining elements , considering as a matrix.

Here, I am selecting the first column to find the determinant because all the entries in that column is $1$. Therefore,

$det\left[\begin{array}{ccc}1&a&bc\\1&b&ca\\1&c&ab\end{array}\right]=1(ab^{2} -ac^{2} )-1(a^{2}b- bc^{2} )+1(ca^{2}-c b^{2} )$

                             $=a(b^{2}- c^{2} )-b(a^{2}- c^{2} )+c(a^{2}- b^{2} )$ . . . . (1)

and

$det\left[\begin{array}{ccc}1&a&a^{2} \\1&b&b^{2} \\1&c&c^{2} \end{array}\right] = 1(bc^{2}-b ^{2}c )-1(ac^{2}- a^{2}c )+1(ab^{2} -a^{2}b )$

                            $=bc^{2} -b^{2} c-ac^{2}+a^{2} c+ab^{2} -a^{2} b\\\\=a(b^{2} -c^{2} )-b(a^{2} -c^{2} )+c(a^{2} -b^{2} )$ . . . . (2)

Since $(1)=(2)$ we can said that the determinant of these  two matrices are equal.

Hence the answer.

thank you