If A and B are square matrices of the same order 3, such that det (A) = 3 and AB = 3I, where I is an identity matrix of order 3. Then the value of det (B) is:

Given:

• A and B are 3×3 square matrices
• det(A) = 3
• AB = 3I, where I is the 3×3 identity matrix

Taking determinant of both sides of AB = 3I:

$\text{det}(AB) = \text{det}(3I)$

Using the property $\text{det}(AB) = \text{det}(A) \times \text{det}(B)$:

$\text{det}(A) \times \text{det}(B) = \text{det}(3I)$

For a 3×3 identity matrix multiplied by scalar 3:

$3I = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$

The determinant of $kI$ for an $n \times n$ matrix is $k^n$.

For the 3×3 case:

$\text{det}(3I) = 3^3 = 27$

Substituting into the equation:

$3 \times \text{det}(B) = 27$

$\text{det}(B) = \dfrac{27}{3}$

$\text{det}(B) = 9$

Therefore, the value of det(B) is 9.