Question:
The value of x, so that the matrix $=\begin{bmatrix}
x+a &b& c \\[0.3em]
a& x+b & c \\[0.3em]
a & b & x+c \end{bmatrix}$ has rank 3 , is
The value of x, so that the matrix $=\begin{bmatrix}
x+a &b& c \\[0.3em]
a& x+b & c \\[0.3em]
a & b & x+c \end{bmatrix}$ has rank 3 , is
Updated On: Jul 7, 2022
- $x\neq 0$
- x = a + b + c
- $x\neq\,0\, $ and $\,x\neq -(a+b+c) $
- $x = 0$ and $ a = a + b+ c .$
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The Correct Option is C
Solution and Explanation
Since rank = 3.
$\therefore$ $\begin{vmatrix}x+a&b&c\\ a&a+b&c\\ a&b&x+c\end{vmatrix} \ne 0 $
Operate $C_1 + C_2 + C_3$
$\begin{vmatrix}x+a+b+c&b&c\\ x+a+b+c&x+b&c\\ x+a+b+c&b&x+c\end{vmatrix} \neq 0 $
$\Rightarrow$ $(x + a + b + c) \begin{vmatrix}1&b&c\\ 1&x+b&c\\ 1&b&x+c\end{vmatrix} \neq 0$
$\Rightarrow$ $x + a +b + c \, \neq 0$ and
$\begin{vmatrix}1&b&c\\ 0&x&0\\ 0&0&x\end{vmatrix} \ne 0 \Rightarrow x^{2} \ne 0 \Rightarrow x \ne 0$
Then $x \neq 0, x + a + b + c \neq 0$
$i.e., x \neq -(a + b + c)$
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
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