Q

# Can someone explain - Matrices and Determinants - JEE Main-2

Let S be the set of all real values of ‘a’ for which the following system of linear equations

is consistent. Then the set S is :

• Option 1)

an empty set

• Option 2)
equal to R
• Option 3)
equal to R − {1}
• Option 4)

equal to  {1}

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N

As we learnt in

Cramer's rule for solving system of linear equations -

When $\Delta =0$  and $\Delta _{1}=\Delta _{2}=\Delta _{3}=0$ ,

then  the system of equations has infinite solutions.

- wherein

$a_{1}x+b_{1}y+c_{1}z=d_{1}$

$a_{2}x+b_{2}y+c_{2}z=d_{2}$

$a_{3}x+b_{3}y+c_{3}z=d_{3}$

and

$\Delta =\begin{vmatrix} a_{1} &b_{1} &c_{1} \\ a_{2} & b_{2} &c_{2} \\ a_{3}&b _{3} & c_{3} \end{vmatrix}$

$\Delta _{1},\Delta _{2},\Delta _{3}$ are obtained by replacing column 1,2,3 of $\Delta$ by $\left ( d_{1},d_{2},d_{3} \right )$  column

$\begin{vmatrix} a & 2 & 5\\ 2 & 1 & 3\\ 0 & 3 & 7 \end{vmatrix}\neq 0$

$a\left ( -2 \right )-2\left ( 14 \right )+5\left ( 6 \right )\neq 0$

$a\neq 1$

Option 1)

an empty set

This option is incorrect.

Option 2)

equal to R

This option is incorrect.

Option 3)

equal to R − {1}

This option is correct.

Option 4)

equal to  {1}

This option is incorrect.

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