Question

Let $A=\begin{pmatrix}1 & 0 & 0 \\\ 0 & 4 & -1 \\\ 0 & 12 & -3\end{pmatrix}$ 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)

4097

Answer 2

A2=⎣⎡​100​0412​0−1−3​⎦⎤​⎣⎡​100​0412​0−1−3​⎦⎤​
=⎣⎡​100​0412​0−1−3​⎦⎤​=A
⇒A3=A4=……=A
(A+I)11=11C0​A11+11C1​A10+….⋅11C10​A+11C11​I
=(11C0​+11C1​+….11C10​)A+I
=(211−1)A+I=2047A+I
∴ Sum of diagonal elements =2047(1+4−3)+3
=4094+3=4097