Question

$A=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right],U_1,U_2$ and $U_3$ are columns matrices satisfying $A{U}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],A{U}_2=\left[\begin{array}{c}2\\ 3\\ 0\end{array}\right],A{U}_3=\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]$ and $U$ is $3\times 3$ matrix whose columns are $U_1,U_2,U_3$ then answer the following question

The value of $\left[\begin{array}{ccc}3& 2& 0\end{array}\right]U\left[\begin{array}{c}3\\ 2\\ 0\end{array}\right]$ is

• A. $\left[5\right]$
• B. $\left[\frac{5}{2}\right]$
• C. $\left[4\right]$
• D. $\left[\frac{3}{2}\right]$

Answer 1

The correct option is A $\left[5\right]$

Let $U_1=(abc{)}^T$

Then, we need to solve three simultaneous linear equations to get a,b,c

$a=1$

$2a+b=0$

$3a+2b+c=0$

On solving, we get $a=1,b=-2,c=1$

So, $U_1=(1-21{)}^T$

Similarly solving for $U_2$ and $U_3$, we get,

$U_2=(2-1-4{)}^T$

$U_3=(2-1-3{)}^T$

$U[320{]}^T=[7-8-5{]}^T$

$\left[320\right][7-8-5{]}^T=[5]$

So, Option (A)