Question

Let $A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 4& -1\\ 0& 12& -3\end{array}\right)$. Then the sum of the diagonal elements of the matrix $(A+I{)}^{11}$ is equal to :

• A. 2050
• B. 4094
• C. 6144
• D. 4097

Answer 1

Answer: (D)

$A^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 4& -1\\ 0& 12& -3\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 4& -1\\ 0& 12& -3\end{array}\right]$
$=\left[\begin{array}{ccc}1& 0& 0\\ 0& 4& -1\\ 0& 12& -3\end{array}\right]=A$
$\Rightarrow A^3=A^4=\dots \dots =A$
$(A+I){}^{11}={}^{11}{C}_0{A}^{11}+{}^{11}{C}_1{A}^{10}+\dots .\cdot {}^{11}{C}_{10}A+{}^{11}{C}_{11}I$
$=\left({}^{11}{C}_0+{}^{11}{C}_1+\dots .{}^{11}{C}_{10}\right)A+I$
$=\left(2^{11}-1\right)A+I=2047A+I$
$\therefore$ Sum of diagonal elements $=2047(1+4-3)+3$
$=4094+3=4097$