Unit 1
Matrices and Determinants
Q1)
A1)
Q2) Add 
.
A2)
A + B = 
Q3) Multiply 
A3)
Then 
Q4) Find the inverse of matrix ‘A’ if-
A4)
Here we have-
Then
And the matrix formed by its co-factors of |A| is-
Then
Therefore-
We know that-
Q5) Find the inverse of matrix ‘A’ by using elementary transformation-
A = 
A5) Write the matrix ‘A’ as-
A = IA
Apply 
, we get
Apply 
Apply 
Apply 
Apply 
So that,
= 
Q6) Find the inverse of matrix ‘A’ by using elementary transformation-
A = 
A6) Write the matrix ‘A’ as-
A = IA
Apply 
Apply 
Apply 
Apply 
So that
= 
Q7) Find the solution of the following homogeneous system of linear equations,
A7) The given system of linear equations can be written in the form of matrix as follows,
Apply the elementary row transformation,
, we get,
, we get
Here r(A) = 4, so that it has trivial solution,
Q8) Find out the value of ‘b’ in the system of homogenenous equations-
2x + y + 2z = 0
x + y + 3z = 0
4x + 3y + bz = 0
Which has
(1) trivial solution
(2) non-trivial solution
A8) (1)
For trivial solution, we already know that the values of x , y and z will be zerp, so that ‘b’ can have any value.
Now for non-trivial solution-
(2)
Convert the system of equations into matrix form-
AX = O
Apply 
respectively , we get the following resultant matrices
For non-trivial solutions , r(A) = 2 < n
b – 8 = 0
b = 8
Q9) Check the consistency and find the values of x , y and z of the following system of linear equations.
2x + 3y + 4z = 11
X + 5y + 7z = 15
3x + 11y + 13z = 25
A9) Re-write the system of equations in augmented matrix form.
C = [A,B]
That will be,
Apply 
Now apply ,
We get,
~
~
Here rank of A = 3
And rank of C = 3, so that the system of equations is consistent,
So that we can can solve the equations as below,
That gives,
x + 5y + 7z = 15 ……………..(1)
y + 10z/7 = 19/7 ………………(2)
4z/7 = 16/7 ………………….(3)
From eq. (3)
z = 4,
from 2,
From eq.(1), we get
x + 5(-3) + 7(4) = 15
That gives,
x = 2
Therefore the values of x , y , z are 2 , -3 , 4 respectively.
Q10) Solve the following equations by using Cramer’s rule-
A10)
Here we have-
And here-
Now by using cramer’s rule-
Q11) Solve the following system of linear equations-
A11)
By using cramer’s rule-
