If x = {bmatrix} 3 & -4 \\ 1 & -1 {bmatrix} then xⁿ is equal to

Correct Answer is : None of these

{bmatrix} 3n & -4n \\ n & -n {bmatrix}

{bmatrix} 2+n & -n \\ n & 5-n {bmatrix}

{bmatrix} 3ⁿ & -4ⁿ \\ 4 & (-1)ⁿ {bmatrix}

Correct Answer is : None of these

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PUNE MCA Exam 2014 Question Paper with answers for online practice

Question : 30 of 58

Marks: +1, -0

If x =

\begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}

x^n

is equal to

Solution:

Concept:

Compute matrix powers to find a pattern for

x^n

Explanation:

Given

x = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}

Compute

x^2 = \begin{bmatrix} 5 & -8 \\ 2 & -3 \end{bmatrix}

and

x^3 = \begin{bmatrix} 7 & -12 \\ 3 & -5 \end{bmatrix}

Observe the pattern:

• Top-left:

3,5,7,\dots

→

2n+1

• Top-right:

-4,-8,-12,\dots

→

-4n

• Bottom-left:

1,2,3,\dots

→

n

• Bottom-right:

-1,-3,-5,\dots

→

1-2n

Thus

x^n = \begin{bmatrix} 2n+1 & -4n \\ n & 1-2n \end{bmatrix}

None of the given options match this form.

Answer:

Option D: None of these

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