Linear Algebra - GeeksforGeeks

21 Jan, 2014
Which one of the following does NOT equal to 
 
 
C 
D 
B 
A 
   
First of all, you should know the basic properties of determinants before approaching For these kind of problems. 1) Applying any row or column transformation does not change the determinant 2) If you interchange any two rows, sign of the determinant will change
 
 
A = | 1 x x^2 | | 1 y y^2 | | 1 z z^2 |
 
To prove option (b)
 
=> Apply column transformation C2 -> C2+C1
 
C3 -> C3+C1
 
=> det(A) = | 1 x+1 x^2+1 | | 1 y+1 y^2+1 | | 1 z+1 z^2+1 |
 
To prove option (c),
 
=> Apply row transformations R1 -> R1-R2
 
R2 -> R2-R3
 
=> det(A) = | 0 x-y x^2-y^2 | | 0 y-z y^2-z^2 | | 1 z z^2 |
 
To prove option (d),
 
=> Apply row transformations R1 -> R1+R2
 
R2 -> R2+R3
 
=> det(A) = | 2 x+y x^2+y^2 | | 2 y+z y^2+z^2 | | 1 z z^2 |
 
A =  1    1
     1   -1

A2 = 2   0
                0   2

A4 = A2 X A2
A4 = 4   0
                0   4

A8 = 16   0
                0    16


A16 = 256   0
                 0    256

A18 = A16 X A2
A18 = 512   0
                 0    512 

A19 = 512   512
                 512  -512


Applying Characteristic polynomial

512-lamda   512
512       -(512+lamda)  =   0

-(512-lamda)(512+lamda) - 512 x 512 = 0

lamda2 = 2 x 5122 
det(A) = -2.
det(A^19) = (det(A))^19 = -2^19 = lambda1*lambda2.
The only viable option is D. 
The Eigen values of a triangular matrix are given by its diagonal entries. We can also calculate (or verify given answers) using characteristic equation obtained by |M - λI| = 0.
 
1-λ    2     3
0     4-λ    7           =       0
0      0     3-λ 
Which means
(1-λ)(4-λ)(3-λ) = 0
Consider the following matrix A = 
If the eigenvalues of A are 4 and 8, then  
x=-3, y=9 
x= -4, y=10 
x=5, y=8 
x=4, y=10 
How many of the following matrices have an eigenvalue 1?
three 
four 
two 
one 
1 0 0 0 1
0 1 1 1 0
0 1 1 1 0
0 1 1 1 0
1 0 0 0 1
Which one of the following statements is TRUE about every  
If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative. 
If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive. 
If the trace of the matrix is positive, all its eigenvalues are positive.  
If the determinant of the matrix is positive, all its eigenvalues are positive. 
The trace of a matrix is the sum of the elements of the principal diagonal of the matrix. Fact - The sum of Eigen values of a matrix is equal to it’s trace. Fact - The product of Eigen values of a matrix is equal to its determinant value. Since it’s given that the trace is positive and the determinant is negative, there must be atleast one negative Eigen value. In general there can be an odd number of negative Eigen values in this case since the determinant value is negative.  
Consider the set H of all 3 × 3 matrices of the type  
where a, b, c, d, e and f are real numbers and abc ≠ 0. Under the matrix multiplication operation, the set H is  
a group 
a monoid but not a group 
a semigroup but not a monoid 
neither a group nor a semigroup 
Because identity matrix is identity & as they define abc != 0, then it is non-singular so inverse is also defined. The set of matrices is the set of Upper triangular matrices(H) of size 3*3 with non-zero determinant. Along with the multiplication operator the set forms an Algebraic Structure since it follows the Closure Property. This is because the product of Two Upper Triangular Matrices is also a Upper Triangular Matrix. The Algebraic Structure also follows the Associative Property since, multiplication of matrices in general follows the Associative Property. Therefore it is a Semi Group. The Algebraic Structure is also a Monoid, since it has an Identity element, which is the Identity Matrix- I3. The Algebraic Structure is a Group since every matrix in H has an inverse, since every matrix in H is non-singular (given in question). The Algebraic Structure is not an Abelian Group since it does not follow the Commutative Property. Therefore Option A is correct. 
Consider the matrix as A
| A | = 2 ( 2 -20 ) +1 ( 3 + 5 ) + 3 ( 12 + 2 )
      = 36 + 8+ 42
     = 86
The determinant value of following matrix is non-zero, therefore we have a unique solution. 
 2   -1   3
 3   -2   5
-1    4   1  
 
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