Question

Consider the system of equations,
$AX=B,$ where
$A=\left[\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}a\\ 0\\ b\end{array}\right]$.

Then which of the following(s) is/are always correct?
$(O$ be the null matrix and $a,b,x,y,z\in R)$

• A. If $\left(\text{adj}A\right)B\ne O$, then the system of equations is always inconsistent.
• B. If $\left(\text{adj}A\right)B\ne O$, then the system of equations may be either consistent or inconsistent depending on $a$ and $b$.
• C. If $\left(\text{adj}A\right)B=O$, then the system of equations is always inconsistent.
• D. If $\left(\text{adj}A\right)B=O$, then the system of equations may be either consistent or inconsistent depending on $a$ and $b$.

Answer 1

Answer: (A), (D)

If $\left(\text{adj}A\right)B=O$, then the system of equations may be either consistent or inconsistent depending on $a$ and $b$.

$A=\left[\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right]$

$|A|=0$

We know that,
$\left(\text{adj}A\right)A=|A|I=A\left(\text{adj}A\right)...\left(1\right)$
$AX=B...\left(2\right)$
Multiplying eqn$\left(2\right)$ by $\text{adj}A$, we get
$\left(\text{adj}A\right)AX=\left(\text{adj}A\right)B$
From eqn$\left(1\right)$,
$|A|IX=\left(\text{adj}A\right)B$

$\text{Case-1}:$
If $\left(\text{adj}A\right)B\ne O$, then the system of equations is inconsistent because
$\text{L.H.S.}=|A|IX=O(\because |A|=0)$
$\text{R.H.S.}=\left(\text{adj}A\right)B\ne O\Rightarrow \text{L.H.S.}\ne \text{R.H.S.}$

$\text{Case-2}:$
If $\left(\text{adj}A\right)B=O$
$\text{L.H.S.}=|A|IX=O(\because |A|=0)$
$\text{R.H.S.}=\left(\text{adj}A\right)B=O\Rightarrow \text{L.H.S.}=\text{R.H.S.}$

The system of equations $AX=B$ may be either consistent or inconsistent depending on whether the system has either infinitely many solutions or no solution (depending on $a$ and $b$)