Numerical Analysis Questions and Answers – Eigen Value and Eigen Vectors

This set of Numerical Analysis Multiple Choice Questions & Answers (MCQs) focuses on “Eigen Value and Eigen Vectors”.

1. What should be the Eigen values for the matrix given below?
\( \begin{bmatrix}
-8 & -6 & -2 \\
-6 & -7 & -4 \\
-2 & -4 & -3
\end{bmatrix}
\)
a) -15, -3, 0
b) -15, -5, 0
c) -25, -3, 0
d) -15, -1, 0
View Answer

Answer: a
Explanation: Given,
|A-λI|=0
\(\begin{bmatrix}
-8-\lambda & -6 & -2 \\
-6 & -7-\lambda & -4 \\
-2 & -4 & -3-\lambda
\end{bmatrix}\) = 0

(-8-λ)((-7-λ) × (-3-λ)-(-4) × (-4))-(-6)((-6)×(-3-λ)-(-4)×(-2))+(-2)((-6)×(-4)-(-7-λ)×(-2))=0
(-8-λ)((21+10λ+λ2)-16)+6((18+6λ)-8)-2(24-(14+2λ)) =0
(-8-λ)(5+10λ+λ2)+6(10+6λ)-2(10-2λ) =0
(-40-85λ-18λ23)+ (60+36λ)-(20-4λ) =0
(-λ3-18λ2 -45λ)=0
-λ (λ+3) (λ+15) =0
λ=0 or (λ+3) =0 or (λ+15) =0
The Eigen-values of the matrix A are given by λ=-15,-3, 0.

2. What should be the Eigen values for the matrix given below?
\( \begin{bmatrix}
6 & -2 & 2 \\
-2 & 3 & -1 \\
2 & -1 & 3
\end{bmatrix}
\)
a) 2 & 8
b) 1 & 8
c) 2 & 1
d) 1 & 2
View Answer

Answer: a
Explanation: Given,
|A-λI|=0
\( \begin{bmatrix}
6 – \lambda & -2 & 2 \\
-2 & 3 – \lambda & -1 \\
2 & -1 & 3 – \lambda
\end{bmatrix}
\)
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= 0

(6-λ)((3-λ) × (3-λ) – (-1) × (-1)) – (-2)((-2) × (3-λ) – ( -1)×2) + 2((-2) × (-1) – (3-λ)×2)=0
(6-λ)((9-6λ + λ2)-1) + 2((-6+2λ) – (-2)) + 2(2-(6-2λ))=0
(6-λ)(8-6λ+λ2) + 2(-4+2λ) + 2(-4+2λ)=0
(48-44λ+12λ23) + (-8+4λ) + (-8+4λ)=0
(-λ3 +12λ2 -36λ+32)=0
-(λ-2)(λ-2)(λ-8)=0
(λ-2)=0 or (λ-2)=0 or (λ-8)=0
The Eigen-values of the matrix A are given by λ=2, 8.

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3. What should be the Eigen values for the matrix given below?
\( \begin{bmatrix}
3 & 2 & 4 \\
2 & 0 & 2 \\
4 & 2 & 3
\end{bmatrix}
\)
a) 1 & 8
b) -1 & 8
c) 2 & 5
d) 5 & 2
View Answer

Answer: b
Explanation: Given,
|A-λI|=0

\( \begin{bmatrix}
3 – \lambda & 2 & 4 \\
2 & -\lambda & 2 \\
4 & 2 & 3 – \lambda
\end{bmatrix}
\) = 0
(3-λ)((-λ) × (3-λ)-2×2) – 2(2×(3-λ)-2×4) + 4(2×2-(-λ)×4)=0
(3-λ)((-3λ+λ2)-4) – 2((6-2λ)-8) + 4(4-(-4λ))=0
(3-λ)(-4-3λ+λ2) – 2(-2-2λ) + 4(4+4λ)=0
(-12-5λ+6λ23) – (-4-4λ) + (16+16λ)=0
(-λ3 +6λ2 +15λ+8)=0
-(λ+1)(λ+1)(λ-8)=0
(λ+1)=0 or(λ+1)=0 or(λ-8)=0
The Eigen-values of the matrix A are given by λ=-1, 8.

4. What is the general form of characteristic equation of matrix?
a) |A-λI| = 0
b) |A-λ| = 0
c) |A-I| = 0
d) |Al-λI| = 0
View Answer

Answer: a
Explanation: The general form of characteristic equation of matrix is |A-λI| = 0. Where ‘A’ is the square matrix & ‘λ’ is any scalar quantity. Roots of characteristics equation are called Eigen values. Eigen values are also called or known as latent roots.
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5. What is the general form for characteristic vectors?
a) (A-λI) X = 0
b) (A-λ) X = 0
c) (A-λI) = 0
d) (Al-λI) X = 0
View Answer

Answer: a
Explanation: The general form of characteristic vectors is (A-λI)X = 0. Where ‘A’ is the square matrix & ‘λ’ is the Eigen value of matrix A and ‘X’ is the Eigen vector corresponding to the Eigen value. Eigen vector is also known as latent vector.

6. What is a diagonal matrix?
a) Eigen values equal to their diagonal elements.
b) Eigen vectors equal to their diagonal elements.
c) Eigen values equal to their diagonal elements being 0.
d) Eigen vectors equal to their diagonal elements being 0.
View Answer

Answer: c
Explanation: Diagonal matrix is a kind of matrices which have their Eigen values equals to their diagonal elements. That is what diagonal matrix is. In addition to being called a diagonal matrix it is furthermore also known to be upper triangle as well as lower triangle.
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7. What should be the Eigen values for the matrix given below?
\( \begin{bmatrix}
6 & 3 \\
4 & 5
\end{bmatrix}
\)
a) 2 & 9
b) 2 & 10
c) 1 & 9
d) 2 & 7
View Answer

Answer: a
Explanation: Given,
|A-λI|=0
\( \begin{bmatrix}
6 – \lambda & 3 \\
4 & 5 – \lambda
\end{bmatrix}
\) = 0

(6-λ) × (5-λ) – 3 × 4=0
(30-11λ+λ2 ) – 12=0
2 -11λ+18)=0
(λ-2)(λ-9)=0
(λ-2)=0 or (λ-9)=0
The Eigen-values of the matrix A are given by λ=2, 9.

8. What should be the Eigen values for the matrix given below?
\( \begin{bmatrix}
-5 & -1 \\
-4 & -2
\end{bmatrix}
\)
a) -6 & -1
b) -12 & -6
c) -1 & -8
d) -2 & -6
View Answer

Answer: a
Explanation: Given,
|A-λI|=0
\( \begin{bmatrix}
-5 – \lambda & -1 \\
-4 & -2 – \lambda
\end{bmatrix}
\) = 0 
(-5-λ)× (-2-λ)-(-1) × (-4) =0
(10+7λ+λ2 )-4=0
2+7λ+6)=0
(λ+1)(λ+6)=0
(λ+1)=0 or (λ+6) =0

The Eigen-values of the matrix A are given by λ=-6, -1.

9. What should be the Eigen vectors of the matrix given below?
\( \begin{bmatrix}
6 & 3 \\
4 & 5
\end{bmatrix}
\)
a) [(-0.75, 1) (1, 1)]
b) [(0.75, 1) (1, 1)]
c) [(-0.75, -1) (1, -1)]
d) [(-0.25, 1) (1, 1)]
View Answer

Answer: a
Explanation: Given,
|A-λI|=0
\( \begin{bmatrix}
6-\lambda & 3 \\
4 & 5-\lambda
\end{bmatrix}
 \) = 0
(6-λ) × (5-λ) – 3 × 4=0
(30-11λ+λ2 ) – 12=0
2 -11λ+18)=0
(λ-2)(λ-9)=0
(λ-2)=0 or (λ-9)=0
The Eigen-values of the matrix A are given by λ=2, 9.
Eigen-vectors for λ=2
A-λI = \( \begin{bmatrix}
6 & 3 \\
4 & 5
\end{bmatrix}\) – 2 \( \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}\)

A-λI = \( \begin{bmatrix}
6 & 3 \\
4 & 5
\end{bmatrix}\) – \( \begin{bmatrix}
2 & 0 \\
0 & 1
\end{bmatrix}\)
= \( \begin{bmatrix}
4 & 3 \\
4 & 3
\end{bmatrix}\)
Now, reduce this matrix
R1←R1/4

\( \begin{bmatrix}
1 & 0.75 \\
4 & 3
\end{bmatrix}\)

=

R2←R2-(4×R1)

\( \begin{bmatrix}
1 & 0.75 \\
0 & 0
\end{bmatrix}\)

=

The system associated with the Eigen-value λ=2

\( (A-2I) \begin{matrix}
x_1\\
x_2 \end{matrix}  = \begin{matrix}
1 & 0.75\\
0 & 0 \end{matrix} 
\begin{matrix}
x_1\\
x_2 \end{matrix} = 
\begin{matrix}
0\\
0 \end{matrix}\)

x1 + 0.75x2 = 0
x1 = -0.75x2
Eigen-vectors corresponding to the Eigen-value λ=2 is

\(v = \begin{matrix}
-0.75x_2\\
x_2
\end{matrix}\)

Let x2 = 1

\( v_1 = \begin{matrix}
-0.75\\
1
\end{matrix}\)

Eigen-vectors for λ=9
A-λI = \( \begin{bmatrix}
6 & 3 \\
4 & 5
\end{bmatrix} – 9
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}\)

A-λI = \( \begin{bmatrix}
6 & 3 \\
4 & 5
\end{bmatrix} –
\begin{bmatrix}
9 & 0 \\
0 & 9
\end{bmatrix} = \begin{bmatrix}
-3 & 3 \\
4 & -4
\end{bmatrix}\)

Interchanging rows R1↔R2

\( \begin{bmatrix}
4 & -4 \\
-3 & 3
\end{bmatrix}\)

R1←R1/4

\( \begin{bmatrix}
1 & -1 \\
-3 & 3
\end{bmatrix}\)
R2←R2+(3×R1)
\( \begin{bmatrix}
1 & -1 \\
0 & 0
\end{bmatrix}\)

The system associated with the Eigen-value λ=9

\( (A-9I) \begin{matrix}
x_1\\
x_2 \end{matrix} = \begin{matrix}
1&-1\\
0&0
\end{matrix}
\begin{matrix} 
x_1\\
x_2 \end{matrix} = \begin{matrix}
0\\
0
\end{matrix}\)

x1 – x2 = 0
x1 = x2
Eigen-vectors corresponding to the Eigen-value λ=9 is

\(v = \begin{matrix}
x_2\\
x_2
\end{matrix}\)

Let x2=1

\(v_2 = \begin{matrix}
1\\
1
\end{matrix}\)

10. Eigen values and Eigen vectors are also called as latent values and latent vectors respectively.
a) True
b) False
View Answer

Answer: a
Explanation: Eigen values are also called as latent roots. Eigen vectors are also called as latent vectors. Furthermore Eigen vectors are also known as characteristic vectors and invariant vectors, similar in the case of Eigen values, they are too known as characteristic roots.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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