Question

Solve `|(4 - x, 4 + x, 4 +  x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)|` = 0

Answer 1

`|(4 - x, 4 + x, 4 +  x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)|` = 0

Put x = 0

`|(4 - 0, 4 + 0, 4 +  0),(4 + 0, 4 - 0, 4 + 0),(4 + 0, 4 + 0, 4 - 0)|` = 0

`|(4, 4, 4),(4, 4, 4),(4, 4, 4)|` = 0

0 = 0

∴ x = 0 satisfies the given equation.

Hence x = 0 is a root of the given equation.

Since three rows are identical, x = 0 is a root of multiplicity 2.

Since the degree of the product of the leading diagonal elements (4 – x)(4 – x)(4 – x) is 3.

There is one more root for the given equation.

`|(4 - x, 4 + x, 4 +  x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)|` = 0

`"C"_1 -> "C"_1 + "C"_2 + "C"_3`

`|(4 - x + 4 + x + 4 + x, 4 + x, 4 + x),(4 + x + 4 - x + 4 + x, 4 - x, 4 + x),(4 + x + 4 + x + 4 - x, 4 + x, 4 - x)|` = 0

`|(12 + x, 4 + x, 4 + x),(12 + x, 4 - x, 4 + x),(12 + x, 4 + x, 4 - x)|` = 0

Put x = – 12

`|(12 - 12, 4 - 12, 4 - 12),(12 - 12, 4 + 12, 4 - 12),(12 - 12, 4 - 12, 4 + 12)|` = 0

`|(0, -8, -8),(0, 16, -8),(0, -8, 16)|` = 0

0 = 0

∴ x = – 12 is a root of the given equation.

Hence, the required roots are x = 0, 0, – 12