If \( A=\begin{bmatrix}1 & -2 & 3\\-4 & 2 & 4\end{bmatrix} \) and \( B=\begin{bmatrix}1 & -2\\3 & -4\\2 & 4\end{bmatrix} \), then \( AB \) is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \begin{bmatrix}1 & 18\\10 & 16\end{bmatrix} \)
A is of order 2 × 3 and B is of order 3 × 2. The number of columns of A equals the number of rows of B, so AB exists and is of order 2 × 2.
Each entry is the row of A times the column of B:
\( \displaystyle (AB)_{11}=(1)(1)+(-2)(3)+(3)(2)=1 \)
\( \displaystyle (AB)_{12}=(1)(-2)+(-2)(-4)+(3)(4)=18 \)
\( \displaystyle (AB)_{21}=(-4)(1)+(2)(3)+(4)(2)=10 \)
\( \displaystyle (AB)_{22}=(-4)(-2)+(2)(-4)+(4)(4)=16 \)
\( \displaystyle AB=\begin{bmatrix}1 & 18\\10 & 16\end{bmatrix} \)
Why the other options are incorrect
Option (1) Not possible — Incorrect. The product is possible, since A has 3 columns and B has 3 rows.
Option (2) \( \begin{bmatrix}1 & -6 & 6\\8 & -8 & 16\end{bmatrix} \) — Incorrect. A 2 × 3 matrix cannot be the product; AB must have as many columns as B, which is 2.
Option (3) \( \begin{bmatrix}9\\19\\15\end{bmatrix} \) — Incorrect. A 3 × 1 matrix cannot be the product; AB must have as many rows as A, which is 2.