176. Consider the matrix A =		50		70
     		70		80

     
     whose eigenvectors corresponding to eigenvalues λ₁ and λ₂ are
     
     respectively. The value of x₁ ^Tx₂ is ________.

1. 1. 2

   
   
   

   2. 1

   
   
   

   3. 0

   
   
   

   4. -1

   
   
   

---

1. View Hint View Answer Discuss in Forum

   A =		50		70
   		70		80

   
   Eigen values of A are λ₁ , λ₂
   λ₁ + λ₂ = 130
   λ₁λ₂ = - 900
   

   Given that X1 =		70
   		λ1	-50

   
   

   X2 =		λ2	-80
   		70

   
   

   X1TX2 = [70λ1 - 50]		λ2	-80
   		70

   
   = 70 λ₂ –5600 + 70 λ₁ – 3500
   = 70 (λ₁ + λ₂) – 9100
   = 70 (130) – 9100 = 0

   Correct Option: C

   A =		50		70
   		70		80

   
   Eigen values of A are λ₁ , λ₂
   λ₁ + λ₂ = 130
   λ₁λ₂ = - 900
   

   Given that X1 =		70
   		λ1	-50

   
   

   X2 =		λ2	-80
   		70

   
   

   X1TX2 = [70λ1 - 50]		λ2	-80
   		70

   
   = 70 λ₂ –5600 + 70 λ₁ – 3500
   = 70 (λ₁ + λ₂) – 9100
   = 70 (130) – 9100 = 0

---

177. The product of eigenvalues of the matrix P is
     

     P =		2		0		1
     		4		-3		3
     		0		2		-1

1. 1. –6

   
   
   

   2. 2

   
   
   

   3. 6

   
   
   

   4. –2

   
   
   

---

1. View Hint View Answer Discuss in Forum

   Product of eigen value = | p |
   

   =		2		0		1		= 2
   		4		-3		3
   		0		2		-1

   Correct Option: B

   Product of eigen value = | p |
   

   =		2		0		1		= 2
   		4		-3		3
   		0		2		-1

---

---

178. The number of linearly independent eigenvectors of matrix
     

     A =		2		0		1		is ____
     		0		2		0
     		0		0		3

1. 1. λ = 2, 2, 3

   
   
   

   2. λ = 2, 3, 3

   
   
   

   3. λ = 2, 3, 2

   
   
   

   4. λ = 3, 2, 3

   
   
   

---

1. View Hint View Answer Discuss in Forum

   Consider A - λI =		2 - λ		1		0
   		0		2 - λ		0
   		0		0		3 - λ

   
   ∴ characteristics equation is | A - λI |
   λ = 2, 2, 3
   There are only two linearly independent eigen vectors.

   Correct Option: A

   Consider A - λI =		2 - λ		1		0
   		0		2 - λ		0
   		0		0		3 - λ

   
   ∴ characteristics equation is | A - λI |
   λ = 2, 2, 3
   There are only two linearly independent eigen vectors.

---

179. The condition for which the eigenvalues of the
     

     matrix A =		2		1		are positive, is
     		1		K

1. 1. k >	1
      	2

   
   
   

   2. k > - 2

   
   
   

   3. k > 0

   
   
   

   4. k < -	1
      	2

   
   
   

---

1. View Hint View Answer Discuss in Forum

   All eigen values of A =		2		1		are position 2 > 0
   		1		k

   
   ∴ 2 × 2 leading minor will be greater then zero.
   

   		2		1		> 0
   		1		k

   
   2 k –1 > 0
   2 K > 1
   

   k >	1	.
   	2

   Correct Option: A

   All eigen values of A =		2		1		are position 2 > 0
   		1		k

   
   ∴ 2 × 2 leading minor will be greater then zero.
   

   		2		1		> 0
   		1		k

   
   2 k –1 > 0
   2 K > 1
   

   k >	1	.
   	2

---

---

180. The lowest eigenvalue of the 2 × 2 matrix
     

     		4		2		is _______.
     		1		3

1. 1. 2

   
   
   

   2. 3

   
   
   

   3. 4

   
   
   

   4. 5

   
   
   

---

1. View Hint View Answer Discuss in Forum

   		4 - λ		2		= 0
   		1		3 - λ

   
   (4 – λ) (3 – λ) –2 = 0
   λ² –7 λ+ 12 – 2 = 0
   λ² –7 λ+ 10 = 0
   λ² –5 λ– 2 λ+ 10 = 0
   λ(λ – 5) – 2 (λ – 5) = 0
   λ₁₂ = 2,5
   The lowest eigenvalve of the matrix is 2.

   Correct Option: A

   		4 - λ		2		= 0
   		1		3 - λ

   
   (4 – λ) (3 – λ) –2 = 0
   λ² –7 λ+ 12 – 2 = 0
   λ² –7 λ+ 10 = 0
   λ² –5 λ– 2 λ+ 10 = 0
   λ(λ – 5) – 2 (λ – 5) = 0
   λ₁₂ = 2,5
   The lowest eigenvalve of the matrix is 2.

---