If x = -9 is a root of A = $\begin{vmatrix}
x & 3 & 7 \\
2 & x & 2 \\
7 & 6 & x \\
\end{vmatrix}$ = 0, then other two root are
- 3,7
- 2,7
- 3,6
- 2,6
The Correct Option is B
Solution and Explanation
We are given the determinant condition:
\(A = \begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix} = 0 \)
Expanding along the first row:
\(A = x(x^2 - 12) - 3(2x - 14) + 7(12 - 7x)\)
Now simplify each term:
\(= x^3 - 12x - 6x + 42 + 84 - 49x\)
Combine like terms:
\(x^3 - 67x + 126 = 0\)
It is given that \(x = 9\) is a root, so \((x + 9)\) is a factor.
Now divide the polynomial \(x^3 - 67x + 126\) by \((x + 9)\) to factorize:
\(x^3 - 67x + 126 = (x + 9)(x^2 - 9x + 14)\)
Further factor the quadratic:
\(x^2 - 9x + 14 = (x - 7)(x - 2)\)
So the complete factorization is:
\((x + 9)(x - 7)(x - 2) = 0\)
Final values of \(x\): \(x = -9, 7, 2\)
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Concepts Used:
Transpose of a Matrix
The matrix acquired by interchanging the rows and columns of the parent matrix is called the Transpose matrix. The transpose matrix is also defined as - “A Matrix which is formed by transposing all the rows of a given matrix into columns and vice-versa.”
The transpose matrix of A is represented by A’. It can be better understood by the given example:
Now, in Matrix A, the number of rows was 4 and the number of columns was 3 but, on taking the transpose of A we acquired A’ having 3 rows and 4 columns. Consequently, the vertical Matrix gets converted into Horizontal Matrix.
Hence, we can say if the matrix before transposing was a vertical matrix, it will be transposed to a horizontal matrix and vice-versa.
Read More: Transpose of a Matrix