78876
If \(A\) is a square matrix such that \(A(\operatorname{adj} A)=\) \(\left[\begin{array}{lll}4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4\end{array}\right]\), then \(\operatorname{det}(\operatorname{adj} A)\) is equal to
78878
If \(a, b, c\) and \(d\) are real number such that \(a^{2}+\) \(b^{2}+c^{2}+d^{2}=1\) and \(A=\left[\begin{array}{rr}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) then \(A^{-}\) 1 equals to
1 \(\left[\begin{array}{rr}a+i b & -c-i d \\ c-i d & a-i b\end{array}\right]\)
2 \(\left[\begin{array}{rr}a-i b & c+i d \\ -c+i d & a+i b\end{array}\right]\)
3 \(\left[\begin{array}{rr}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\)
4 \(\left[\begin{array}{ccc} a+i b & c+i d \\ c-i d & a-i b\end{array}\right]\)
Explanation:
(C) : Given, \(A=\left[\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) \(\mathrm{A}^{-1}=\frac{\operatorname{adj}(\mathrm{A})}{|\mathrm{A}|}\) \(|A|=(a+i b)(a-i b)-(c+i d)(-c+i d)\) \(|A|=a^{2}+b^{2}-\left[-d^{2}-c^{2}\right]=a^{2}+b^{2}+c^{2}+d^{2}\) \(|\mathrm{A}|=1\) \(\operatorname{Adj} A=\left[\begin{array}{cc}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\) \(A^{-1}=\left[\begin{array}{cc}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\)
78876
If \(A\) is a square matrix such that \(A(\operatorname{adj} A)=\) \(\left[\begin{array}{lll}4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4\end{array}\right]\), then \(\operatorname{det}(\operatorname{adj} A)\) is equal to
78878
If \(a, b, c\) and \(d\) are real number such that \(a^{2}+\) \(b^{2}+c^{2}+d^{2}=1\) and \(A=\left[\begin{array}{rr}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) then \(A^{-}\) 1 equals to
1 \(\left[\begin{array}{rr}a+i b & -c-i d \\ c-i d & a-i b\end{array}\right]\)
2 \(\left[\begin{array}{rr}a-i b & c+i d \\ -c+i d & a+i b\end{array}\right]\)
3 \(\left[\begin{array}{rr}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\)
4 \(\left[\begin{array}{ccc} a+i b & c+i d \\ c-i d & a-i b\end{array}\right]\)
Explanation:
(C) : Given, \(A=\left[\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) \(\mathrm{A}^{-1}=\frac{\operatorname{adj}(\mathrm{A})}{|\mathrm{A}|}\) \(|A|=(a+i b)(a-i b)-(c+i d)(-c+i d)\) \(|A|=a^{2}+b^{2}-\left[-d^{2}-c^{2}\right]=a^{2}+b^{2}+c^{2}+d^{2}\) \(|\mathrm{A}|=1\) \(\operatorname{Adj} A=\left[\begin{array}{cc}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\) \(A^{-1}=\left[\begin{array}{cc}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\)
78876
If \(A\) is a square matrix such that \(A(\operatorname{adj} A)=\) \(\left[\begin{array}{lll}4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4\end{array}\right]\), then \(\operatorname{det}(\operatorname{adj} A)\) is equal to
78878
If \(a, b, c\) and \(d\) are real number such that \(a^{2}+\) \(b^{2}+c^{2}+d^{2}=1\) and \(A=\left[\begin{array}{rr}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) then \(A^{-}\) 1 equals to
1 \(\left[\begin{array}{rr}a+i b & -c-i d \\ c-i d & a-i b\end{array}\right]\)
2 \(\left[\begin{array}{rr}a-i b & c+i d \\ -c+i d & a+i b\end{array}\right]\)
3 \(\left[\begin{array}{rr}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\)
4 \(\left[\begin{array}{ccc} a+i b & c+i d \\ c-i d & a-i b\end{array}\right]\)
Explanation:
(C) : Given, \(A=\left[\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) \(\mathrm{A}^{-1}=\frac{\operatorname{adj}(\mathrm{A})}{|\mathrm{A}|}\) \(|A|=(a+i b)(a-i b)-(c+i d)(-c+i d)\) \(|A|=a^{2}+b^{2}-\left[-d^{2}-c^{2}\right]=a^{2}+b^{2}+c^{2}+d^{2}\) \(|\mathrm{A}|=1\) \(\operatorname{Adj} A=\left[\begin{array}{cc}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\) \(A^{-1}=\left[\begin{array}{cc}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\)
78876
If \(A\) is a square matrix such that \(A(\operatorname{adj} A)=\) \(\left[\begin{array}{lll}4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4\end{array}\right]\), then \(\operatorname{det}(\operatorname{adj} A)\) is equal to
78878
If \(a, b, c\) and \(d\) are real number such that \(a^{2}+\) \(b^{2}+c^{2}+d^{2}=1\) and \(A=\left[\begin{array}{rr}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) then \(A^{-}\) 1 equals to
1 \(\left[\begin{array}{rr}a+i b & -c-i d \\ c-i d & a-i b\end{array}\right]\)
2 \(\left[\begin{array}{rr}a-i b & c+i d \\ -c+i d & a+i b\end{array}\right]\)
3 \(\left[\begin{array}{rr}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\)
4 \(\left[\begin{array}{ccc} a+i b & c+i d \\ c-i d & a-i b\end{array}\right]\)
Explanation:
(C) : Given, \(A=\left[\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right]\) \(\mathrm{A}^{-1}=\frac{\operatorname{adj}(\mathrm{A})}{|\mathrm{A}|}\) \(|A|=(a+i b)(a-i b)-(c+i d)(-c+i d)\) \(|A|=a^{2}+b^{2}-\left[-d^{2}-c^{2}\right]=a^{2}+b^{2}+c^{2}+d^{2}\) \(|\mathrm{A}|=1\) \(\operatorname{Adj} A=\left[\begin{array}{cc}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\) \(A^{-1}=\left[\begin{array}{cc}a-i b & -c-i d \\ c-i d & a+i b\end{array}\right]\)
