Question

If a, b, c are all different and the equations
$ax + {a^2}y + ({a^3} + 1) = 0$
$bx + {b^2}y + ({b^3} + 1) = 0$
$cx + {c^2}y + ({c^3} + 1) = 0$
are consistent, then prove that abc + 1 = 0.

Answer 1

We will use the concept of determinant to prove the given condition. The determinant of a matrix A is denoted by det(A), detA, or |A|. If the given sets of equations are consistent then for that case the determinant of the given equations is equal to zero. Arranging the given equations in the matrix form and putting the determinant equals to zero, we can get the required answer.

Complete step by step answer:
Now, from the question we have,
We will arrange the given sets of equation in matrix form
\left( {\begin{array}{*{20}{c}} a&{{a^2}}&{{a^3} + 1} \\\ b&{{b^2}}&{{b^3} + 1} \\\ c&{{c^2}}&{{c^3} + 1} \end{array}} \right)
Now for the consistency of the above equation, D = 0
det\left( {\begin{array}{*{20}{c}} a&{{a^2}}&{{a^3} + 1} \\\ b&{{b^2}}&{{b^3} + 1} \\\ c&{{c^2}}&{{c^3} + 1} \end{array}} \right) = 0
expanding along column 1, we get
1 + abc = 0

NOTE: Matrix is a rectangular array or table of numbers, symbols or expressions arranged in rows and columns. For example, the matrix given here is a matrix of dimension 2$\times$2 (2 by 2) because there are two rows and two columns \left( {\begin{array}{*{20}{c}} 1&0 \\\ 3&1 \end{array}} \right). If there are two matrices of the same dimension (same number of rows and same number of columns), then the matrices can be added and subtracted. But, for the matrix multiplication the number of columns in the first matrix is equal to the number of rows in the second matrix.
For matrix addition, number of rows = number of columns
For matrix multiplication, $(m \times n) = (n \times p)$ resulting in $(n \times p)$ matrix.
If this condition is satisfied then only the product of two matrices is possible, else it is not possible. Matrices are generally denoted using capital letters such as A, B, C, etc.