Explanation

A matrix is symmetric if it is equal to its transpose. That means:

$$A = A^T$$

For this 3×3 matrix:

$$A = begin{bmatrix} 5 & x & 3 \ x & 7 & 2 \ 3 & 2 & 9 \ end{bmatrix}$$

We compare element aij​ with $a_{ji}$ The (1,2) element is $x$, and the (2,1) element is also $x$, so they're already equal.
Now, look at element (1,3), which is 3, and element (3,1), which is also 3 — so they match.
Now check (2,3) = 2 and (3,2) = 2 — they also match.

So, the matrix is symmetric if and only if:

• $x=x$ (always true)

• All off-diagonal symmetric pairs match

Given that, the matrix is symmetric already, but if in the original problem the matrix looked like this:

$$begin{bmatrix} 5 & x & 3 \ 11 & 7 & 2 \ 3 & 2 & 9 \ end{bmatrix}$$

Then, to make it symmetric:

• The element at (1,2) must equal the element at (2,1)

• So: $$x=11
  

Explanation

The Inductive Method starts with specific examples or observations and leads to the formation of a general principle or rule.

This approach is often used in language learning, where students observe language patterns before being taught the rules.

Explanation

 Universal Grammar, proposed by Noam Chomsky, argues that the ability to acquire grammar is hard-wired into the brain and that all languages share a common structural basis.

Explanation

The Deductive Method starts with presenting a general principle or theory, followed by specific examples or applications.

It’s often used in grammar teaching where rules are introduced first.

Explanation

An effective teacher often needs authority to maintain classroom control, but also flexibility to adapt to different teaching situations and students’ needs.

Explanation

The Inquiry Method encourages students to actively explore and ask questions, which can lead to deep learning, but it is often considered time-consuming because it requires more time for investigation, discussion, and discovery.

Explanation

The primary focus of Phrase Structure Rules in grammar is syntactic structure. 

Phrase Structure Rules (PSRs) define how phrases and sentences are constructed by specifying how words and phrases can be combined to form larger units like noun phrases, verb phrases, and entire sentences.

They essentially govern the hierarchical structure of a sentence, outlining the grammatical relationships between different components.