# Linear Algebra Questions and Answers – Eigenvalues and Vectors of a Matrix

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This set of Linear Algebra Multiple Choice Questions & Answers (MCQs) focuses on “Eigenvalues and Vectors of a Matrix”.

1. Find the Eigen values for the following 2×2 matrix.
A=$$\begin{bmatrix}1&8\\2&1\end{bmatrix}$$.
a) -3
b) 2
c) 6
d) 4

Explanation: We know that for any given matrix
[A-λI]X=0 and |A-λI|=0
[A-λI]=$$\begin{bmatrix}1&8\\2&1\end{bmatrix} -\lambda \begin{bmatrix}1&0\\0&1\end{bmatrix}$$
[A-λI]=$$\begin{bmatrix}1-\lambda &8\\2&1-\lambda \end{bmatrix}$$
|A-λI|=(1-λ)(1-λ)-16=0
(1-λ)2=16
(1-λ)=±4
λ=-3 or λ=5.

2. Find the Eigenvalue for the given matrix.
A=$$\begin{bmatrix}4&1&3\\1&3&1\\2&0&5\end{bmatrix}$$.
a) 13
b) -3
c) 7.1
d) 8.3

Explanation: We know that for any given matrix
[A-λI]X=0 and |A-λI|=0
[A-λI]=$$\begin{bmatrix}4&1&3\\1&3&1\\2&0&5\end{bmatrix}-\lambda \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$
The characteristic polynomial is given by-
λ3-(Sum of diagonal elements) λ2+(Sum of minor of diagonal element)λ-|A|=0
λ3-12λ2+40λ-39=0
λ=7.1 or λ=3.

3. Find the Eigen vector for value of λ=-2 for the given matrix.
A=$$\begin{bmatrix}3&5\\3&1\end{bmatrix}$$.
a) $$\begin{bmatrix}0\\-1\end{bmatrix}$$
b) $$\begin{bmatrix}1\\-1\end{bmatrix}$$
c) $$\begin{bmatrix}-1\\-1\end{bmatrix}$$
d) $$\begin{bmatrix}1\\0\end{bmatrix}$$

Explanation: We know that for any given matrix
[A-λI]X=0 and |A-λI|=0
Given that, λ=-2
[A-λI]=$$\begin{bmatrix}3&5\\3&1\end{bmatrix}-(-2)\begin{bmatrix}1&0\\0&1\end{bmatrix}$$
[A-λI]=$$\begin{bmatrix}3+2&5\\3&1+2\end{bmatrix}$$
[A-λI]=$$\begin{bmatrix}5&5\\3&3\end{bmatrix}$$
Since, [A-λI]X=0
$$\begin{bmatrix}5&5\\3&3\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix}$$
Thus,
5x+5y=0 and 3x+3y=0
Let x=t,
Then, y=-t
X = $$\begin{bmatrix}t\\-t\end{bmatrix} = \begin{bmatrix}1\\-1\end{bmatrix}$$
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4. Find the Eigen vector for value of λ=3 for the given matrix.
A=$$\begin{bmatrix}3&10&5\\-2&-3&-4\\3&5&7\end{bmatrix}$$.
a) $$\begin{bmatrix}-1\\-1\\2\end{bmatrix}$$
b) $$\begin{bmatrix}-1\\1\\2\end{bmatrix}$$
c) $$\begin{bmatrix}-1\\-1\\-2\end{bmatrix}$$
d) $$\begin{bmatrix}-1\\-2\\2\end{bmatrix}$$

Explanation: We know that for any given matrix
[A-λI]X=0 and |A-λI|=0
Given that, λ=3
[A-λI]=$$\begin{bmatrix}3&10&5\\-2&-3&-4\\3&5&7\end{bmatrix}-(3)\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$
[A-λI]=$$\begin{bmatrix}3-3&10&5\\-2&-3-3&-4\\3&5&7-3\end{bmatrix}$$
[A-λI]=$$\begin{bmatrix}0&10&5\\-2&-6&-4\\3&5&4\end{bmatrix}$$
Since, [A-λI]X=0
$$\begin{bmatrix}0&10&5\\-2&-6&-4\\3&5&4\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$$
Using Row transformation

(1) Interchanging R1 and R2/2
$$\begin{bmatrix}-1&-3&-2\\0&10&5\\3&5&4\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$$
(2) R3=R3+3R1
$$\begin{bmatrix}0&10&5\\0&10&5\\0&-4&-2\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$$
(3) R2=R2/5 and R3=R3+2R2
$$\begin{bmatrix}-1&-3&-2\\0&2&1\\0&0&0\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$$
Thus,
-x-3y-2z=0 and 2y+z=0
Let z=t,then y=$$\frac{-t}{2}$$ and x=$$\frac{-t}{2}$$

X = $$\begin{bmatrix}-1\\-1\\2\end{bmatrix}$$

5. Find the Eigen value and the Eigen Vector for the given matrix.
A=$$\begin{bmatrix}3&4&2\\1&6&2\\1&4&4\end{bmatrix}$$.
a) 3, $$\begin{bmatrix}1\\1\\1\end{bmatrix}$$
b) 9, $$\begin{bmatrix}1\\1\\1\end{bmatrix}$$
c) 9, $$\begin{bmatrix}1\\0\\1\end{bmatrix}$$
d) 2, $$\begin{bmatrix}1\\0\\1\end{bmatrix}$$

Explanation: We know that for any given matrix
[A-λI]X=0 and |A-λI|=0
[A-λI]=$$\begin{bmatrix}3&4&2\\1&6&2\\1&4&4\end{bmatrix}-\lambda \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$
The characteristic polynomial is given by-
λ3-(Sum of diagonal elements) λ2+(Sum of minor of diagonal element)λ-|A|=0

λ3-13λ2+40λ-36=0
λ=9 or λ=2
For λ=9,
[A-λI]=$$\begin{bmatrix}3&4&2\\1&6&2\\1&4&4\end{bmatrix}-9\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$
[A-λI]=$$\begin{bmatrix}3-9&4&2\\1&6-9&2\\1&4&4-9\end{bmatrix}$$
[A-λI]=$$\begin{bmatrix}-6&4&2\\1&-3&2\\1&4&-5\end{bmatrix}$$
[A-λI]X=0
$$\begin{bmatrix}-6&4&2\\1&-3&2\\1&4&-5\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$$

(1) Interchanging R1 and R2
$$\begin{bmatrix}1&-3&2\\-6&4&2\\1&4&-5\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$$

(2) R2=R2+6R1 and R3=R3-R1
$$\begin{bmatrix}1&-3&2\\0&-14&14\\0&7&-7\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$$

(3) R3=R3+R2/2
$$\begin{bmatrix}1&-3&2\\0&-14&14\\0&0&0\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}$$

-14y+14z=0
x-3y+2z=0
Let z=t
Then, y=t and x=t
X=$$\begin{bmatrix}1\\1\\1\end{bmatrix}$$.

Sanfoundry Global Education & Learning Series – Linear Algebra.