# What is the determinant of the square matrix A(x) [(2x+1) -2] [(2x+1) (2x-1)] ?

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For a square matrix

|a b |

|c d |

the determinant is given by (ad - bc)

In the problem the matrix given is:

|(2x+1) - 2 |

|(2x+1) (2x-1)|

The determinant of the matrix is: (2x + 1)(2x - 1) - (-2)*(2x + 1)

=> (2x + 1)(2x - 1) + 2*(2x + 1)

4x^2 - 1 + 4x + 2

=> 4x^2 + 4x + 1

**The required determinant is 4x^2 + 4x + 1**

Since the given matrix is a square matrix, we may evaluate it's determinant.

Det A = a11*a22 - a12*a21

The determinant is the difference between the product of the terms located on the first diagonal (1st term from the 1st row* the 2nd term from the 2nd row) and the product of the terms located on the 2nd diagonal (2nd term from the 1st row *the 1st term from the 2nd row)

Let's identify the terms: a11 = (2x+1) ; a12 = (-2) ; a21 = (2x+1) and a22 = (2x-1).

Det A(x) = (2x+1)*(2x-1) - (-2)*(2x+1)

Det A = (2x+1)*(2x - 1 + 2)

Det A = (2x+1)*(2x + 1)

Det A = (2x+1)^2

**The requested determinant of the square matrix A(x) is Det A = (2x+1)^2.**