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Question:
The system $2\,x+3 y+z=5,3\,x+y+5\,z=7$
and $x+4\,y-2\,z=3$ has
The system $2\,x+3 y+z=5,3\,x+y+5\,z=7$
and $x+4\,y-2\,z=3$ has
Updated On: Aug 15, 2024
- unique solution
- finite number of solution
- infinite solutions
- no solution
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The Correct Option is D
Solution and Explanation
The given system can be written as $A X=B$, where
$A =\begin{bmatrix}2 & 3 & 1 \\ 3 & 1 & 5 \\ 1 & 4 & -2\end{bmatrix}, X=\begin{bmatrix}x \\ y \\ z\end{bmatrix}, B=\begin{bmatrix}5 \\ 7 \\ 3\end{bmatrix}$
$\therefore |A| =\begin{vmatrix}2 & 3 & 1 \\ 3 & 1 & 5 \\ 1 & 4 & -2\end{vmatrix}$
$=2(-2-20)-3(-6-5)+1(12-1) $
$=2(-22)-3(-11)+1(11)$
$=-44+33+11=0 $
$ \because |A| =0 $
Hence, there exist no solution.
$A =\begin{bmatrix}2 & 3 & 1 \\ 3 & 1 & 5 \\ 1 & 4 & -2\end{bmatrix}, X=\begin{bmatrix}x \\ y \\ z\end{bmatrix}, B=\begin{bmatrix}5 \\ 7 \\ 3\end{bmatrix}$
$\therefore |A| =\begin{vmatrix}2 & 3 & 1 \\ 3 & 1 & 5 \\ 1 & 4 & -2\end{vmatrix}$
$=2(-2-20)-3(-6-5)+1(12-1) $
$=2(-22)-3(-11)+1(11)$
$=-44+33+11=0 $
$ \because |A| =0 $
Hence, there exist no solution.
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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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