If \(A=\left[\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) and…
If \(A=\left[\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) and \(A^{-1}=\left[\begin{array}{cc}a+i b & -c-i d \\ -c+i d & a-i b\end{array}\right]\). Find \(\left(a^2+b^2+c^2+d^2\right)\).
1
-1
\(i\)
\(-i\)
Solution
Given matrix
\(A=\left[\begin{array}{cc}
a+i b & c+i d \\
-c+i d & a-i b
\end{array}\right]\)
So, \(\quad A^{-1}=\frac{1}{|A|}\left[\begin{array}{cc}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\)
\(\begin{aligned}
& =\frac{1}{a^2+b^2+c^2+d^2}\left[\begin{array}{cc}
a-i b & -c-i d \\
c-i d & a+i b
\end{array}\right] \\
& =\left[\begin{array}{cc}
a+i b & -c-i d \\
-c+i d & a-i b
\end{array}\right] \quad \text{(given)}
\end{aligned}\)
\(\therefore \quad b=0\) and \(c=d=0\)
and \(a^2+b^2+c^2+d^2=1\)