The values of $\alpha$ for which the system of equation x + y + z = 1, x + 2y + 4z = $\alpha$, x + 4y + 10z = $\alpha^2$ is consistent are given by
- 1, -2
- -1, 2
- 44563
- 44562
The Correct Option is C
Solution and Explanation
Given the system:
\(\begin{vmatrix} 1 & 1 & 1 & : & 1 \\ 1 & 2 & 4 & : & \alpha \\ 1 & 4 & 10 & : & \alpha^2 \end{vmatrix}\)
Apply row operations:
\(R_2 \rightarrow R_2 - R_1,\quad R_3 \rightarrow R_3 - R_1\)
\(\begin{vmatrix} 1 & 1 & 1 & : & 1 \\ 0 & 1 & 3 & : & \alpha - 1 \\ 0 & 3 & 9 & : & \alpha^2 - 1 \end{vmatrix}\)
Now eliminate using \(R_3 \rightarrow R_3 - 3R_2\):
\(\begin{vmatrix} 1 & 1 & 1 & : & 1 \\ 0 & 1 & 3 & : & \alpha - 1 \\ 0 & 0 & 0 & : & \alpha^2 - 3\alpha + 2 \end{vmatrix}\)
For the system to be consistent, the last equation must satisfy:
\(\alpha^2 - 3\alpha + 2 = 0\)
Factorising:
\((\alpha - 1)(\alpha - 2) = 0\)
Hence,
\(\alpha = 1 \text{ or } 2\)
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Concepts Used:
Matrix Transformation
The numbers or functions that are kept in a matrix are termed the elements or the entries of the matrix.
Transpose Matrix:
The matrix acquired by interchanging the rows and columns of the parent matrix is termed the Transpose matrix. The definition of a transpose matrix goes as follows - “A Matrix which is devised by turning all the rows of a given matrix into columns and vice-versa.”