Question

Consider a 2 $\times$ 2 matrix M = $\left[v_1{v}_2\right]$ where, $v_1$ and $v_2$ are the colum vectors. Suppose $M^{-1}=\left[\begin{array}{c}{U}_1^T\\ U_2^T\end{array}\right]$, where $U_1^T$ and $U_2^T$ are the row vectors. Consider the following statements :

Statement I : $U_1^T.V_1=1$ and $U_2^T.V_2=1$
Statement II : $U_1^T.V_2=1$ and $U_2^T.V_1=0$

• A. Statement 1 is true and statement 2 is false
• B. Both the statements are false
• C. Statement 2 is true and statement 1 false
• D. Both the statements are true.

Answer 1

The correct option is D Both the statements are true.

Let M = $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ =$\left[\begin{array}{cc}{V}_1& V_2\end{array}\right]$,
$V_1=\left[\begin{array}{c}a\\ c\end{array}\right],V_2=\left[\begin{array}{c}b\\ d\end{array}\right]$

$M^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]=\left[\begin{array}{cc}{U}_1^T& \\ U_2^T& \end{array}\right]$

$U_1^T=\left[\begin{array}{cc}\frac{d}{ad-bc}& \frac{-b}{ad-bc}\end{array}\right]$
$U_2^T=\left[\begin{array}{cc}\frac{-c}{ad-bc}& \frac{a}{ad-bc}\end{array}\right]$
Checking given statement
$U_1^T{V}_1=\left[\begin{array}{cc}\frac{d}{ad-bc}& \frac{-b}{ad-bc}\end{array}\right]\left[\begin{array}{c}a\\ c\end{array}\right]=1$
$=\frac{ad-bc}{ad-bc}=1$
$U_2^T{V}_2=\left[\begin{array}{cc}\frac{-c}{ad-bc}& \frac{a}{ad-bc}\end{array}\right]\left[\begin{array}{c}b\\ d\end{array}\right]$
$=\frac{-bc+ad}{ad-bc}=1$
So, statement 1 is true.

Statement -II

$U_1^T{V}_2=\left[\begin{array}{cc}\frac{d}{ad-bc}& \frac{-b}{ad-bc}\end{array}\right]\left[\begin{array}{c}b\\ d\end{array}\right]$
$=\frac{bd}{ad-bc}-\frac{bd}{ad-bc}=0$
$U_2^T{V}_1=\left[\begin{array}{cc}\frac{-c}{ad-bc}& \frac{a}{ad-bc}\end{array}\right]\left[\begin{array}{c}a\\ c\end{array}\right]$
$=\frac{-ac}{ad-bc}+\frac{ac}{ad-bc}=0$

Statement II is true