Question

Real matrices $[A{]}_{3\times 1}$ , $[B{]}_{3\times 3}$ ,$[C{]}_{3\times 5}$, $[D{]}_{5\times 3}$, $[E{]}_{5\times 5}$ and $[F{]}_{5\times 1}$ are given. matrices [B] and [E] are symmetric.

Following statements are made with respect to these matrices.

1. Matrix product $[F{]}^T$ $[C{]}^T$ [B] [C] [F] is a scalar.

2. Matrix product $[F{]}^T$ [F] [D] is always symmetric

With reference to above statements, which of the following applies?

• A. Statement 1 is true but 2 is false
• B. Statement 1 is false but 2 is true
• C. Both the Statements are true
• D. Both the Statements are false

Answer 1

The correct option is A Statement 1 is true but 2 is false

Given real matrices

$[A{]}_{3\times 1}$, $[B{]}_{3\times 3}$ ,$[C{]}_{3\times 5}$, $[D{]}_{5\times 3}$ , $[E{]}_{5\times 5}$, $[F{]}_{5\times 1}$

Also, $[B{]}^T$ = [B] & $[E{]}^T$ = [E]

(1) Order $[F{]}^T$ $[C{]}^T$ [B] [C] [F]

$\Rightarrow$ (1 $\times$ 5) (5 $\times$ 3) (3 $\times$ 3) (3 $\times$ 5) (5 $\times$1)

$\Rightarrow$ Scalar matrix of order (1$\times$ 1) i.e, (1) is true

(2) Order of $[F{]}^T$ [F] [D]

$\Rightarrow$ (1 $\times$ 5) (5 $\times$ 1) (5 $\times$ 3) Not possible to multiply so (2) is false