Matrices & DeterminantsmediumFree

Matrices & Determinants — JEE Maths practice question

JEE Maths question with a full step-by-step solution.

Question

For a matrix

A = \begin{pmatrix}1 & 2r-1\\ 0 & 1\end{pmatrix}

, the value of

\displaystyle\prod_{r=1}^{50}\begin{pmatrix}1 & 2r-1\\ 0 & 1\end{pmatrix}

is equal to:

\begin{pmatrix}1 & 100\\ 0 & 1\end{pmatrix}

\begin{pmatrix}1 & 4950\\ 0 & 1\end{pmatrix}

\begin{pmatrix}1 & 5050\\ 0 & 1\end{pmatrix}

\begin{pmatrix}1 & 2500\\ 0 & 1\end{pmatrix}

correct

Solution

Step 1: Establish the multiplication rule

\begin{pmatrix}1 & p\\ 0 & 1\end{pmatrix}\begin{pmatrix}1 & q\\ 0 & 1\end{pmatrix} = \begin{pmatrix}1 & p+q\\ 0 & 1\end{pmatrix}

By induction, the product of

n

such matrices accumulates the upper-right entries:

\prod_{r=1}^{50}\begin{pmatrix}1 & 2r-1\\ 0 & 1\end{pmatrix} = \begin{pmatrix}1 & \displaystyle\sum_{r=1}^{50}(2r-1)\\ 0 & 1\end{pmatrix}

Step 2: Evaluate the sum

\sum_{r=1}^{50}(2r-1) = 1+3+5+\cdots+99 = 50^2 = 2500

\prod_{r=1}^{50}\begin{pmatrix}1 & 2r-1\\ 0 & 1\end{pmatrix} = \begin{pmatrix}1 & 2500\\ 0 & 1\end{pmatrix}

Answer: (4)

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