Unit-1

Matrices and determinants

Q1) Define matrices.

A1)

A matrix is a rectangular arrangement of the numbers.

These numbers inside the matrix are known as elements of the matrix.

A matrix ‘A’ is expressed as-

The vertical elements are called columns and the horizontal elements are rows of the matrix.

Q2) What are the triangular matrices.

A2) Triangular matrix-

If every element above or below the leading diagonal of a square matrix is zero, then the matrix is known as a triangular matrix.

There are two types of triangular matrices-

(a) Lower triangular matrix-

If all the elements below the leading diagonal of a square matrix are zero, then it is called lower triangular matrix.

Example:

A =

(b) Upper triangular matrix-

If all the elements above the leading diagonal of a square matrix are zero, then it is called lower triangular matrix.

Example-

A =

Q3) check whether the following matrix A is symmetric or not?

A = 

A3)

As we know that if the transpose of the given matrix is same as the matrix itself then the matrix is called symmetric matrix.

So that, first we will find its transpose,

Transpose of matrix A ,

Here,

A =

The matrix A is symmetric.

Q4) What is Hermitian matrix?

A4)

A square matrix A = is said to be hermitian matrix if every element of A is equal to conjugate complex j-ith element of A.

It means,

For example:

Necessary and sufficient condition for a matrix A to be hermitian –

                                            A = (͞A)’

Q5) Express the matrix A as sum of hermitian and skew-hermitian matrix where

A5)

Let A =

Therefore    and

        Let

Again

Hence P is a hermitian matrix.

Let 

 Again

Hence Q is a skew- hermitian matrix.

We Check

  P +Q=

Hence proved.

Q6) whether the following matrix A is symmetric or not?

      A =

A6)

This is not a skew symmetric matrix, because the transpose of matrix A is not equals to  -A.

-A = A’

Q7) Define unitary matrix.

A7)

A square matrix A is said to be unitary matrix if the product of the transpose of the conjugate of matrix A and matrix itself  is an identity matrix.

Such that,

( ͞A)’. A = I

For example:

and its

Then  (͞A)’ . A = I

So that we can say that matrix A is said to be a unitary matrix.

Q8) Add .

A8)

 A + B =

Q9)   then AB?

A9)

Q10) If A = then find |A|.

A10)

As we know that-

Then-

Q11) Expand the determinant:

A11)

As we know

Then,                         

Q12) Find the minors and cofactors of the first row of the determinant.

A12)

The minor of element 2 will be,

Delete the corresponding row and column of element 2,

We get,

Which is equivalent to, 1 × 7 - 0 × 2 = 7 – 0 = 7

Similarly the minor of element 3 will be,  

4× 7 - 0× 6 = 28 – 0 = 28

Minor of element 5,  

4 × 2 - 1× 6 = 8 – 6 = 2

The cofactors of 2, 3 and 5 will be,

Q13) What are the properties of determinants.

A13)

Properties of determinants-

(1) If the rows are interchanged into columns or columns into rows then the value of determinants does not change.

Let us consider the following determinant:

(2) The sign of the value of determinant changes when two rows or two columns are interchanged.

Interchange the first two rows of the following, we get

(3) If two rows or two columns are identical the the value of determinant will be zero.

Let, the determinant has first two identical rows,

As we know that if we interchange the first two rows then the sign of the value of the determinant will be changed, so that

Hence proved

(4) if the element of any row of a determinant be each multiplied by the same number then the determinant multiplied by the same number,

Q14) Solve-

A14)

Given

Apply-

We get-

Q15) Show that the points given below are collinear-

A15)

First we need to find the area of these points and if the area is zero then we can say that these are collinear points-

So that-

We know that area enclosed by three points-

Apply-

So that these points are collinear.

Q16) Find the inverse of matrix ‘A’ if-

A16) Here we have-

Then

And the matrix formed by its co-factors of |A| is-

Then

Therefore-

We know that-

Q17) Find the inverse of matrix ‘A’ by using elementary transformation-

A =

A17)

Write the matrix ‘A’ as-  

A = IA

Apply , we get

Apply

Apply

Apply

Apply

So that,

  =

Q18) check whether the following system of linear equations is consistent of not.

2x + 6y = -11

6x + 20y – 6z = -3

6y – 18z = -1

A18)

Write the above system of linear equations in augmented matrix form,

Apply  , we get

Apply

Here the rank of C is 3 and the rank of A is 2

Therefore both ranks are not equal. So that the given system of linear equations is not consistent.

Q19) Solve the following equations by using Cramer’s rule-

A19)

Here we have-

And here-

Now by using cramer’s rule-

Q20) Solve the following system of linear equations-

A20)
 

By using cramer’s rule-