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COMEDK 2025 Afternoon Solved Paper

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Question : 145 of 180

Marks: +1, -0

If

A(\operatorname{adj} A)=\left[\begin{array}{lll}5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{array}\right]

, then the value of

| A | + | \operatorname{adj} A |

is equal to :

5
25
125
30

Solution:

Concept:

For a square matrix

A

, the product

A(\operatorname{adj} A) = |A| I

, and

|\operatorname{adj} A| = |A|^{n-1}

for an

n \times n

matrix.

Explanation:

Given

A(\operatorname{adj} A) = \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix} = 5I_3

.

Since

A(\operatorname{adj} A) = |A| I_3

, we compare to get

|A| = 5

.

Here

n = 3

, so

|\operatorname{adj} A| = |A|^{n-1} = 5^{3-1} = 5^2 = 25

.

Therefore,

|A| + |\operatorname{adj} A| = 5 + 25 = 30

.

Answer:

Option D (30).

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