A and B are square matrices of order 3. In A the sum of the elements of each row is 2, and in B the sum of the elements of each row is 3. The sum of all the elements of the matrix \(AB^2\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 54
Let \(U = \begin{bmatrix}1\\1\\1\end{bmatrix}\). The product of a matrix with U is the column of its row sums, so
\(AU = 2U\)
\(BU = 3U\)
Using the associative law,
\(B^2U = B(BU) = B(3U) = 3(BU) = 9U\)
\(AB^2U = A(9U) = 9(AU) = 18U\)
So each of the 3 rows of \(AB^2\) has sum 18, and the sum of all its elements is
\(3 \times 18 = 54\)
Why the other options are incorrect
Option (1) 18 — Incorrect. It is the sum of one row of \(AB^2\), not of all three rows.
Option (3) 36 — Incorrect. It takes the row sum of \(B^2\) as \(3 + 3 = 6\) instead of \(3 \times 3 = 9\).
Option (4) 108 — Incorrect. It is twice the correct sum; each row sum of \(AB^2\) is \(2 \times 9 = 18\), not 36.