Assertion (A):If $\mathrm{A}=\left[\begin{array}{cc}2& 1+2\mathrm{i}\\ 1-2\mathrm{i}& 7\end{array}\right]$ then det(A) is real.

Reason (R):If $\mathrm{A}=\left[\begin{array}{cc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}\end{array}\right], {\mathrm{a}}_{\mathrm{ij}}$ being complex numbers, then |A | is always real.

Complete Solution:

Assertion (A)

To find the determinant of A:

det(A) = (2)(7) - (1 + 2i)(1 - 2i)

Expanding (1 + 2i)(1 - 2i):

(1 + 2i)(1 - 2i) = 1 - (2i)² = 1 - (-4) = 5

Substituting back:

det(A) = 14 - 5 = 9

Hence, the determinant is 9, which is real. Assertion (A) is true.

Reason (R)

The determinant of a 2x2 matrix with complex entries is:

det(A) = a11 * a22 - a12 * a21

This is not always real unless specific conditions, like symmetry or conjugate pairs, are met. Generally, det(A) can be complex for arbitrary complex numbers.

Hence, Reason (R) is false.

Final Answer

The determinant of the given matrix A is real, so Assertion (A) is true. The reasoning provided in Reason (R) is incorrect because |A| is not always real for complex matrices.

The correct option is: A is true but R is false.