Matrices, Transpose of a Matrix, Scalar Multiplication of a Matrix

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Transpose of a Matrix

Associated with each given matrix there exists another matrix called its transpose. The transpose of a given matrix A is formed by interchanging its rows and columns and is denoted by or , e.g. if

, then

In general, If is an matrix, then the transpose of is the matrix; and element of element of

Symmetric Matrix:

A square matrix is said to be a symmetric matrix if .

For example,

If , then

Since , is a symmetric matrix.

Skew – Symmetric Matrix:

A square matrix is said to be a skew symmetric if , i.e. for all and .

For example,

If , then

But , which is the same as

Hence, is a skew symmetric matrix.

Scalar Multiplication of a Matrix:

Let us consider the following situation:

The marks obtained by three students in English, Hindi, and Mathematics are as follows:

The Marks Obtained by Three Students
The marks obtained by three students

English

Hindi

Mathematics

Elizabeth

20

10

15

Usha

22

25

27

Shabnam

17

25

21

It is also given that these marks are out of in each case. In matrix form, the above information can be written as

(It is understood that rows correspond to the names and columns correspond to the subjects)

If the maximum marks are doubled in each case, then the marks obtained by these girls will also be doubled. In matrix form, the new marks can be given as:

which is equal to

So, we write that

Now consider another matrix

Let us see what happens, when we multiply the matrix by

i.e.

For example,

If then

When

So,

Thus, if ,then

Example:

If , find (i) (ii)

Solution:

Here,

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