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PUNE MCA Exam 2014 Question Paper with answers for online practice

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Question : 1 of 58

Marks: +1, -0

If A =

\begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}

then find the value of

\mid

^{2009-5}

^{2008}

∣ is

-6
-4
-5
None of these

Solution:

Concept:

Use determinant properties:

\det(AB)=\det A \cdot \det B

and

\det(cA)=c^n\det A

for an

n\times n

matrix.

Explanation:

Step 1: Factor the expression:

A^{2009} - 5A^{2008} = A^{2008}(A - 5I)

.

Step 2: Take determinant:

\det(A^{2009} - 5A^{2008}) = \det(A^{2008}) \cdot \det(A - 5I)

.

Step 3:

\det(A^{2008}) = (\det A)^{2008}

. Compute

\det A = (1)(5) - (2)(3) = 5-6 = -1

. So

(\det A)^{2008} = (-1)^{2008} = 1

.

Step 4: Compute

A - 5I = \begin{bmatrix} 1-5 & 2 \\ 3 & 5-5 \end{bmatrix} = \begin{bmatrix} -4 & 2 \\ 3 & 0 \end{bmatrix}

. Its determinant is

(-4)(0) - (2)(3) = 0-6 = -6

.

Step 5: Multiply:

1 \cdot (-6) = -6

.

Answer:

-6

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