Matrices & Determinants
Trace
MMTS_Full_Test_13
Grade 12
Question:
The set of natural numbers is divided into arrays of rows and columns in the form of matrices as $A_1=[1]$, $A_2=\begin{bmatrix}2&3\\4&5\end{bmatrix}$, $A_3=\begin{bmatrix}6&7&8\\9&10&11\\12&13&14\end{bmatrix}$ and so on. Let the trace of $A_{10}$ be $\lambda$. Find unit digit of $\lambda$.
Step-by-Step Solution
Key Concept: Find first element of $A_n$ and trace (sum of diagonal)
Last element of $A_9$ is $1+2^2+\cdots+9^2-1=285$. First of $A_{10}$ is 286. Trace $=286+287+288+\cdots$ (diagonal, 10 terms with step $n+1=11$): $286+297+308+\cdots$... Sum $= 10\cdot 286+11(0+1+\cdots+9)=2860+495=3355$. Unit digit $=5$.
Correct Answer: 5