Consider the matricesA =−1 2 −22 2 −1, B =3 94 5−3 2 , and C =1 −23 −4Compute the following:(i) AT C2(ii) detC−1
CC−1
T(iii)2AT − BT

(i) To compute AT C^2, we first need to find the transpose of A (AT) and the square of C (C^2).

Matrix A is given as: A = [ -1 2 -2 2 2 -1 ]

The transpose of A (AT) is obtained by interchanging the rows and columns of A. So, AT is: AT = [ -1 2 2 2 -2 -1 ]

Matrix C is given as: C = [ 1 -2 3 -4 ]

The square of C (C^2) is obtained by multiplying C with itself. So, C^2 is: C^2 = [ 11 + -23 1*-2 + -2*-4 31 + -43 3*-2 + -4*-4 ] = [ -5 6 -9 10 ]

Now, we can compute AT C^2 by multiplying AT with C^2: AT C^2 = [ -1*-5 + 2*-9 -16 + 210 2*-5 + 2*-9 26 + 210 -2*-5 + -1*-9 -26 + -110 ] = [ -13 14 -14 16 1 -16 ]

(ii) To compute det(C^-1 (CC^-1)^T), we first need to find the inverse of C (C^-1) and the transpose of CC^-1 (CC^-1)^T.

The inverse of a 2x2 matrix [ a b ; c d ] is given by 1/(ad-bc) * [ d -b ; -c a ]. So, C^-1 is: C^-1 = 1/(1*-4 - -2*3) * [ -4 2 -3 1 ] = 1/(-4 - -6) * [ -4 2 -3 1 ] = [ 2 -1 1.5 -0.5 ]

Now, we can compute CC^-1: CC^-1 = [ 12 + -21.5 1*-1 + -2*-0.5 32 + -41.5 3*-1 + -4*-0.5 ] = [ 1 0 0 1 ]

The transpose of CC^-1 (CC^-1)^T is obtained by interchanging the rows and columns of CC^-1. So, (CC^-1)^T is: (CC^-1)^T = [ 1 0 0 1 ]

Finally, we can compute det(C^-1 (CC^-1)^T) by multiplying C^-1 with (CC^-1)^T and finding the determinant of the resulting matrix. The determinant of a 2x2 matrix [ a b ; c d ] is given by ad - bc. So, det(C^-1 (CC^-1)^T) is: det(C^-1 (CC^-1)^T) = det([ 21 + -10 20 + -11 1.51 + -0.50 1.50 + -0.51 ]) = det([ 2 -1 1.5 -0.5 ]) = 2*-0.5 - -1*1.5 = -1 - -1.5 = 0.5

(iii) To compute (2AT - B)^T, we first need to find 2AT and subtract B from it.

2AT is obtained by multiplying each element of AT by 2: 2AT = [ -2 4 4 4 -4 -2 ]

Matrix B is given as: B = [ 3 9 4 5 -3 2 ]

Now, we can compute 2AT - B: 2AT - B = [ -2 - 3 4 - 9 4 - 4 4 - 5 -4 - -3 -2 - 2 ] = [ -5 -5 0 -1 1 -4 ]

The transpose of 2AT - B (2AT - B)^T is obtained by interchanging the rows and columns of 2AT - B. So, (2AT - B)^T is: (2AT - B)^T = [ -5 0 1 -5 -1 -4 ]