Unit 1
Matrices and determinants
Q1) If A = then show that
(i) is hermitian matrix.
(ii) is skew-hermitian matrix.
A1)
Given A =
Then
Let
Also
Hence P is a Hermitian matrix.
Let
Also
Hence Q is a skew-hermitian matrix.
Q2) Add .
A2)
A + B =
Q3) If A = then find |A|.
A3)
As we know that-
Then-
Q4) Find out the determinant of the following matrix A.
A4)
By the rule of determinants-
Q5) Expand the determinant:
A5) As we know
Then,
Q6) Find the minors and cofactors of the first row of the determinant.
A6) (1) 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,
Q7) Show that the points given below are collinear-
A7)
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.
Q8) Find the inverse of matrix ‘A’ if-
A8)
Here we have-
Then
And the matrix formed by its co-factors of |A| is-
Then
Therefore-
We know that-
Q9) Find the inverse of matrix ‘A’ by using elementary transformation-
A =
A9) Write the matrix ‘A’ as-
A = IA
Apply
Apply
Apply
Apply
So that
=
Q10) Let .
Solve the following homogeneous system of linear equations
Ax= 0
Explain why there are no solutions, an infinite number of solutions, or exactly one solution.
A10)
Note that any homogeneous system is consistent and has at least the trivial solution.
Transform the coefficient matrix to the triangular or row echelon form.
The rank of A equals 3. Therefore, there are no free variables and the system
Has only the trivial solution: