NCERT Class 12-Mathematics: Exemplar Chapter –3 Matrices Part 11

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Question 28:

If , they verify that:

(i)

(ii)

Answer:

We have,

And

Also

Hence proved.

(ii)

And

Hence proved.

Question 29:

Show that and are both symmetric matrices for any matrix A.

Answer:

We must understand,

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, because equal matrices have equal dimensions, only square matrices can be symmetric.

And we know that, transpose of AB is given by

Using this result, take transpose of .

Transpose of

Using, transpose of

And also,

So,

Since,

This means, is symmetric matrix for any matrix A.

Now, take transpose of .

Transpose of

Since,

This means, is symmetric matrix for any matrix A.

Thus, and are symmetric matrix for any matrix A.

Question 30:

Let A and B be square matrices of the order . Is ? Give reasons.

Answer:

We are given that,

A and B are square matrices of the order .

We need to check whether is true or not.

Take .

each, A and B can be multiplied; A and B be any matrices of order ]

[; as A can be multiplied with itself and B can be multiplied by itself]

So, note that, is possible.

But this is possible if and only if .

And is always true whenever A and B are square matrices of any order. And for ,