IPMAT Indore 2025 (SA) - If A = bmatrix 2 & n \\ 4 & 1 bmatrix such that A^3 = 27 bmatrix 4 & q \\ p & r bmatrix, then p + q + r equals _________ | PYQs + Solutions

IPMAT Indore 2025

Modern Math

Matrices & Determinants

Medium

If $A = \begin{bmatrix} 2 & n \\ 4 & 1 \end{bmatrix}$ such that $A^3 = 27 \begin{bmatrix} 4 & q \\ p & r \end{bmatrix}$ , then $p + q + r$ equals _________

Entered answer:

✅ Correct Answer: 12

A^2 = \begin{bmatrix} 2 & n \\ 4 & 1 \end{bmatrix} \begin{bmatrix} 2 & n \\ 4 & 1 \end{bmatrix} = \begin{bmatrix} 4 + 4n & 3n \\ 12 & 4n + 1 \end{bmatrix}

A^3 = \begin{bmatrix} 4 + 4n & 3n \\ 12 & 4n + 1 \end{bmatrix} \begin{bmatrix} 2 & n \\ 4 & 1 \end{bmatrix} = \begin{bmatrix} 8 + 20n & 7n + 4n^2 \\ 28 + 16n & 16n + 1 \end{bmatrix}

Comparing the above

A^3

value with the given data:

27\begin{bmatrix} 4 & q \\ p & r \end{bmatrix} = \begin{bmatrix} 108 & 27q \\ 27p & 27r \end{bmatrix}=\boxed{\begin{bmatrix} 8 + 20n & 7n + 4n^2 \\ 28 + 16n & 16n + 1 \end{bmatrix}}

8 + 20n = 108 \implies n = 5

7n + 4n^2 = 27q \implies 35 + 100 = 27q \implies q = 5

28 + 16n = 27p \implies 28 + 80 = 27p \implies p = 4

16n + 1 = 27r \implies 80 + 1 = 27r \implies r = 3

Therefore,

p + q + r = 4 + 5 + 3 = 12