If \(P\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}\), then matrix P is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\begin{bmatrix}1&-2\\2&0\end{bmatrix}\)
P multiplies a 2 × 3 matrix and gives a 2 × 3 matrix, so P has 2 rows and 2 columns (order 2 × 2). Let
\( \displaystyle P = \begin{bmatrix}p&q\\r&s\end{bmatrix}\)
The first row of the product is \(p(1, 2, 3) + q(4, 5, 6)\), and it must equal \((-7, -8, -9)\):
\(p + 4q = -7\)
\(2p + 5q = -8\)
Doubling the first equation and subtracting the second:
\(3q = -6\), so \(q = -2\) and \(p = 1\)
The third entry checks: \(3p + 6q = 3 - 12 = -9\).
The second row of the product is \(r(1, 2, 3) + s(4, 5, 6)\), and it must equal \((2, 4, 6)\):
\(r + 4s = 2\)
\(2r + 5s = 4\)
\(3s = 0\), so \(s = 0\) and \(r = 2\)
The third entry checks: \(3r + 6s = 6\).
\( \displaystyle P = \begin{bmatrix}1&-2\\2&0\end{bmatrix} \)
Shortcut: The first row of the product depends only on the first row of P, and its first entry is \(p + 4q\). Options (1) and (4) have first row (1, 2), giving \(1 + 8 = 9\); Option (3) has (1, 0), giving 1. Only Option (2), with (1, −2), gives \(1 - 8 = -7\).
NCERT (2026-27 edition):
Matrices
