Assume P, Q, R and W are matrices of order $3 × 3, a × 4,b × c$ and $d × a$ respectively. If $PQ + WR$ is well defined, then the value of $ab + cd$ is:

21

The correct answer is Option (3) → 21

Given matrices:

$P$: $3 \times 3$, $Q$: $a \times 4$, $R$: $b \times c$, $W$: $d \times a$

Expression: $PQ + WR$ must be defined → dimensions must match

1. $PQ$ is defined: columns of $P$ = rows of $Q$ → $3 = a \Rightarrow a = 3$

Then $PQ$ has dimensions $3 \times 4$

2. $WR$ is defined and must have same dimensions as $PQ$ → $W$ is $d \times a = d \times 3$, $R$ is $b \times c$

For $WR$ to be defined: columns of $W$ = rows of $R$ → $3 = b \Rightarrow b = 3$

Then $WR$ has dimensions $d \times c = 3 \times 4 \Rightarrow d = 3$, $c = 4$

Compute $ab + cd = a*b + c*d = 3*3 + 4*3 = 9 + 12 = 21$