لخّصلي

1.The factor 2323 accounts for the scalar multiplication of the 3x3 matrix A2A2 by 2, again assuming a 3x3 matrix for the cubing to make sense.The matrix dimension isn't specified, but the equation suggests a cube operation, implying a manipulation based on properties of determinants.The determinant of a scalar multiplied by a matrix is the scalar raised to the power of the matrix's dimension (here assumed to be nn) times the determinant of the matrix (|cA|=cn|A||cA|=cn|A|).A-1A-1 is the inverse of AA. The determinant of a product of matrices equals the product of their determinants (|AB|=|A||B||AB|=|A||B|).This implies:

2A2A-12A2A-1 involves scaling A2A2 (which is AA multiplied by itself) by 2, then multiplying by A-1A-1.3.

1. Determinant of AA (∣A∣∣A∣):

You're given that the determinant of matrix AA is 2x2+14x+242x2+14x+24, and then this is evaluated at x=1x=1, giving:
∣A∣=2(1)2+14(1)+24=2+14+24=40
∣A∣=2(1)2+14(1)+24=2+14+24=40
2. Determinant involving the product of 3A3A and A−1A−1:

The expression ∣3AA−1∣=33×∣A∣⋅∣A−1∣∣
∣​3AA−1∣

∣​=33×∣A∣⋅∣A−1∣ is being evaluated. Let's clarify this step-by-step:

3A3A is simply the matrix AA scaled by 3.

A−1A−1 is the inverse of AA.

The determinant of a product of matrices equals the product of their determinants (∣AB∣=∣A∣∣B∣∣AB∣=∣A∣∣B∣).

The determinant of a scalar multiplied by a matrix is the scalar raised to the power of the matrix's dimension (here assumed to be nn) times the determinant of the matrix (∣cA∣=cn∣A∣∣cA∣=cn∣A∣).

Since A−1A−1 is the inverse of AA, ∣A−1∣=1/∣A∣∣A−1∣=1/∣A∣.

The matrix dimension isn't specified, but the equation suggests a cube operation, implying a manipulation based on properties of determinants.

Given that, 3333 comes from the scaling factor 33 applied to all elements of AA, assuming AA is a 3x3 matrix (since it's cubed, 3n3n where n=3n=3). The product ∣A∣⋅∣A−1∣∣A∣⋅∣A−1∣ equals 1 because the determinant of a matrix multiplied by the determinant of its inverse is 1.

So, ∣3AA−1∣=27∣
∣​3AA−1∣

∣​=27, showing a simplified evaluation without needing the actual determinant of AA or A−1A−1.
3. Determinant involving 2A2A−12A2A−1:

The expression ∣2A2A−1∣=23×∣A2∣⋅1/∣A∣∣
∣​2A2A−1∣

∣​=23×∣A2∣⋅1/∣A∣ is given. This implies:

2A2A−12A2A−1 involves scaling A2A2 (which is AA multiplied by itself) by 2, then multiplying by A−1A−1.

The determinant properties used here include the product rule and the scaling rule mentioned before.

∣A2∣∣A2∣ means you take the determinant of AA squared, which intuitively can be seen as ∣A∣2∣A∣2.

The factor 2323 accounts for the scalar multiplication of the 3x3 matrix A2A2 by 2, again assuming a 3x3 matrix for the cubing to make sense.

The term 1/∣A∣1/∣A∣ adjusts for the inclusion of A−1A−1.

Simplifying gives 8×∣A∣8×∣A∣. Given ∣A∣=40∣A∣=40, as previously calculated, 8×40=3208×40=320.