Let \(P=\left[\begin{array}{lll}1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1\end{array}\right]\) and \(Q=\left[q_{i j}\right]\) be two \(3 \times 3\) matrices such that \(Q-P^{5}=I_{3}\). Then \(\frac{q_{21}+q_{31}}{q_{32}}\) is equal to:

JEE Main · 2019 · 12 Jan 2019 · Shift 1 · Matrices

Let

P = 139013001

and

Q = [q_{ij}]

be two

3 \times 3

matrices such that

Q - P^{5} = I_{3}

. Then

\frac{q _{21} + q _{31}}{q _{32}}

is equal to:

Answer: (A)

The matrix

P

is lower-triangular with

1

s on the diagonal, and raising it to powers keeps that shape while the sub-diagonal entries grow in a clear pattern. Computing the powers gives

P^{5} = 1151350115001

. Since

Q = P^{5} + I_{3}

, adding the identity changes only the diagonal, so the relevant entries are

q_{21} = 15

q_{31} = 135

and

q_{32} = 15

. Therefore

\frac{q _{21} + q _{31}}{q _{32}} = \frac{15 + 135}{15} = 10

JEE Main · 2019 · 12 Jan 2019 · Shift 1 · Single correct option