If $\begin{bmatrix}1&x&1\end{bmatrix} \begin{bmatrix}1&3&2\\ 2&5&1\\ 15&3&2\end{bmatrix}\begin{bmatrix}1\\ 2\\ x\end{bmatrix} = 0 $, then x can be
- -2
- 2
- 14
- -14
The Correct Option is D
Solution and Explanation
$\Rightarrow \begin{bmatrix}1+2 x+15 \\ 3+5 x+3 \\ 2+x+2\end{bmatrix}\begin{bmatrix}1 \\ 2 \\ x\end{bmatrix}=0$
$\Rightarrow \begin{bmatrix}2 x+16 \\ 5 x+6 \\ x+4\end{bmatrix}\begin{bmatrix}1 \\ 2 \\ x\end{bmatrix}=0$
$\Rightarrow 2 x+16+2(5 x+6)+x(x+4)=0$
$\Rightarrow \, 2 x+16+10 x+12+x^{2}+4 x=0$
$\Rightarrow \, x^{2}+16 x+28=0$
$\Rightarrow \, x^{2}+2 x+14 x+28=0$
$\Rightarrow \, x(x+2)+14(x+2)=0$
$\Rightarrow \,(x+2)(x+14)=0$
$\Rightarrow \,x=-2,-14$
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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.
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