Let A be an invertible square matrix of order 3 × 3. Then | ({adj} A) · A | is

Correct Answer is : ∣A∣3| A | ³∣A∣3

Correct Answer is : ∣A∣3| A | ³∣A∣3

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GUJCET 2025 Math Paper

Question : 16 of 40

Marks: +1, -0

Let

A

3 \times 3

| (\text{adj} A) \cdot A |

Solution:

Concept:

For an invertible matrix

A

, the product of its adjoint and itself is related to its determinant and the identity matrix.

Explanation:

Use the property

(\mathrm{adj}\, A) \cdot A = |A| I

Take determinant on both sides:

|(\mathrm{adj}\, A) \cdot A| = ||A|I|

Since

A

3 \times 3

, the identity matrix

I

is also

3 \times 3

For a scalar

k

and an

n \times n

matrix,

|kI| = k^n |I|

Here

k = |A|

and

n = 3

, so

||A|I| = |A|^3 |I|

Also

|I| = 1

Therefore,

|(\mathrm{adj}\, A) \cdot A| = |A|^3

Answer:

|A|^3

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