The following system of linear equations has
Option: 1 infinitely many solutions, satisfying
Option: 2 infinitely many solutions, satisfying
Option: 3 no solution
Option: 4 only the trivial solution.
System of Homogeneous linear equations 
If ? ≠ 0, then x= 0, y = 0, z = 0 is the only solution of the above system. This solution is also known as a trivial solution.
If ? = 0, at least one of x, y and z are nonzero. This solution is called a nontrivial solution.
Explanation: using equation (ii) and (iii), we have
This is the condition for a system have Nontrivial solution.

so infinite nontrivial solution exist
now equation (1) + 3 equation (3)
10x  20z = 0
x = 2z
Correct Option 2
View Full Answer(1)Let If then :
Option: 1
Option: 2
Option: 3
Option: 4
Elementary row operations 
Elementary row operations
Row transformation: Following three types of operation (Transformation) on the rows of a given matrix are known as elementary row operation (transformation).
i) Interchange of i^{th} row with j^{th} row, this operation is denoted by
ii) The multiplication of ith row by a constant k (k≠0) is denoted by
iii) The addition of ith row to the elements of jth row multiplied by constant k (k≠0) is denoted by
In the same way, threecolumn operations can also be defined too.

Correct Option (3)
View Full Answer(1)If for some in R, the intersection of the following three planes is aline in , then is equal to :
Option: 1
Option: 2
Option: 3
Option: 4
Cramer’s law 
Cramer’s law for the system of equations in two variables :
We can observe that first row in the numerator of x is of constants and 2nd row in the numerator is of constants, and the denominator is of the coefficient of variables.
We can follow this analogy for the system of equations of 3 variable where third in the numerator of the value of z will be of constant and denominator will be formed by the value of coefficients of the variables.
i) If ? ≠ 0, then the system of equations has a unique finite solution and so equations are consistent, and solutions are
ii) If ? = 0, and any of
Then the system of equations is inconsistent and hence no solution exists.
iii) If all then
System of equations is consistent and dependent and it has an infinite number of solution.

View Full Answer(1)
The number of all matrices A, with enteries from the set such that the sum of the diagonal elements of is 3, is
Option: 1 672
Option: 2 512
Option: 31024
Option: 4 256
Let matrix A be
So out of 9 elements, 3 elements must be equal to 1 or −1, and the rest elements must be 0.
Possible cases
View Full Answer(1)For which of the following ordered pairs , the system of linear equations is inconsistent ?
Option: 1
Option: 2
Option: 3
Option: 4
Solution of System of Linear Equations Using Matrix Method 
let us consider n linear equations in n unknowns, given as below
The above system of equations can be written in matrix form as
Premultiplying equation AX=B by A^{1}, we get
A^{1}(AX) = A^{1}B ⇒ (A^{1}A)X = A^{1}B
⇒ IX = A^{1}B
⇒ X = A^{1}B
⇒
Types of equation :
System of equations is nonhomogenous:
If A ≠ 0, then the system of equations is consistent and has a unique solution X = A^{1}B
If A = 0 and (adj A)·B ≠ 0, then the system of equations is inconsistent and has no solution.
If A = 0 and (adj A)·B = 0, then the system of equations is consistent and has infinite number of solutions.
System of equations is homogenous:
If A ≠ 0, then the system of equations has only trivial solution and it has one solution.
If A = 0 then the system of equations has nontrivial solution and it has an infinite number of solution.
If number of equation < number of unknown then it has nontrivial solution.

Hence, it has infinitely many solutions
For non infinite solution
Correct Option (4)
View Full Answer(1)If the system of linear equations 2x+2ay+az=0 2x+3by+bz=0, 2x+4cy+cz=0, where are nonzero and distinct ; has a nonzero solution, then :
Option: 1
Option: 2 are in AP
Option: 3 are in A.P.
Option: 4 are in G.P.
Cramer’s law 
Cramer’s law for the system of equations in two variables :
We can observe that first row in the numerator of x is of constants and 2nd row in the numerator is of constants, and the denominator is of the coefficient of variables.
We can follow this analogy for the system of equations of 3 variable where third in the numerator of the value of z will be of constant and denominator will be formed by the value of coefficients of the variables.
i) If ? ≠ 0, then the system of equations has a unique finite solution and so equations are consistent, and solutions are
ii) If ? = 0, and any of
Then the system of equations is inconsistent and hence no solution exists.
iii) If all then
System of equations is consistent and dependent and it has an infinite number of solution.

For non zero solutions D = 0
on solving
Correct Option (3)
View Full Answer(1)The system of linear equations has :
Option: 1 no solution when
Option: 2 infinitely many solutions when
Option: 3 no solution when
Option: 4 a unique solution when
Cramer’s law 
Cramer’s law for the system of equations in two variables :
We can observe that first row in the numerator of x is of constants and 2nd row in the numerator is of constants, and the denominator is of the coefficient of variables.
We can follow this analogy for the system of equations of 3 variable where third in the numerator of the value of z will be of constant and denominator will be formed by the value of coefficients of the variables.
i) If ? ≠ 0, then the system of equations has a unique finite solution and so equations are consistent, and solutions are
ii) If ? = 0, and any of
Then the system of equations is inconsistent and hence no solution exists.
iii) If all then
System of equations is consistent and dependent and it has an infinite number of solution.

D = ( + 8) ( 2 – ) for = 2
= 5[18 – 10] – 2 [48 – 50] + 2 (16 – 30]
= 40 + 4 – 28 0 No solutions for = 2
Correct Option (1)
View Full Answer(1)If and , then is equal to :
Option: 1
Option: 5
Option: 9
Option: 13
Inverse of a Matrix 
A nonsingular square matrix “A” is said to be invertible if there exists a nonsingular square matrix B such that AB = I = BA, (all matrix are of the same order, they must be for this), then B is called inverse of matrix A.
Inverse of 2 x 2 matrix

Multiplication of two matrices 
Matrix multiplication:
Two matrices A and B are conformable for the product AB if the number of columns in A and the number of rows in B is equal. Otherwise, these two matrices will be nonconformable for matrix multiplication. So on that basis,
i) AB is defined only if col(A) = row(B)
ii) BA is defined only if col(B) = row(A)
If
For examples

On comparing we get
Correct Option (2)
View Full Answer(1)If the system of linear equations, x+y+z=6 x+2y+3z=10 has more than two solutions, then is equal to _______.
Option: 1 13
Option: 2 17
Option: 3 21
Option: 4 25
Cramer’s law 
Cramer’s law for the system of equations in two variables :
We can observe that first row in the numerator of x is of constants and 2nd row in the numerator is of constants, and the denominator is of the coefficient of variables.
We can follow this analogy for the system of equations of 3 variable where third in the numerator of the value of z will be of constant and denominator will be formed by the value of coefficients of the variables.
i) If ? ≠ 0, then the system of equations has a unique finite solution and so equations are consistent, and solutions are
ii) If ? = 0, and any of
Then the system of equations is inconsistent and hence no solution exists.
iii) If all then
System of equations is consistent and dependent and it has an infinite number of solution.

x + y + z = 6 …….. (1)
x + 2y + 3z = 10 …….. (2)
3x + 2y + z = …….. (3)
from (1) and (2)
if z = 0 x + y = 6 and x + 2y = 10
y = 4, x = 2
(2, 4, 0)
if y = 0 x + z = 6 and x + 3z = 10
z = 2 and x = 4
(4, 0, 2)
so, 3x + 2y + z = must pass through (2, 4, 0) and (4, 0, 2)
= 14
and 12 + 2 =
= 1
= 13
View Full Answer(1)Let and be two real matrices such that where, i, j=1,2,3. if the determinant of B is 81, then the determinant of A is :
Option: 1
Option: 2
Option: 3
Option: 4
Matrices, order of matrices, row and column matrix 
A set of numbers (real or complex) or objects or symbols arranged in form of a rectangular array having m rows and n columns and bounded by brackets [?] is called matrix of order m × n, read as m by n matrix.
E.g for m = 2 and n =3, we have order of this matrix is 2×3
The m by n matrix is represented as :
This representation can be represented in a more compact form as
Where represents element of i^{th} row and j^{th} column and i = 1,2,...,m; j = 1,2,...,n.
For example, to locate the entry in matrix A identified as a_{ij}, we look for the entry in row i, column j. In matrix A, shown below, the entry in row 2, column 3 is a_{23}.
Matrix is only a representation of the symbol, number or object. It does not have any value. Usually, a matrix denoted by capital letters.

Properties of Determinants  Part 2 
Property 5
If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the determinant is multiplied by k.
For example
Note:
By this property, we can take out any common factor from any one row or any one column of a given determinant.
If corresponding elements of any two rows (or columns) of a determinant are proportional (in the same ratio), then the determinant value is zero.

Taking Common from and from
Taking Common from and from
Correct Option (A)
View Full Answer(1)The following system of linear equations
If for some in R, the intersection of the following three planes
The number of all matrices A, with enteries from the set
For which of the following ordered pairs , the system of linear equations
The system of linear equations
If the system of linear equations, x+y+z=6 x+2y+3z=10 has more than two solutions, then