Final answer:
To find the determinant of matrix A, use a 2x2 matrix formula, det(A) = (a*d) - (b*c), where A = [5e^(3t) 2e^(4t); -4e^(3t) -3e^(4t)]. The determinant of A is -7e^(7t). To find the matrix of cofactors, calculate the cofactor of each element using the formula C_ij = (-1)^(i+j) * det(M_ij), where M_ij is the submatrix obtained by removing the i-th row and j-th column from A. The matrix of cofactors is C = [(-3e^(4t)) (2e^(4t)); (4e^(3t)) (5e^(3t))].
Step-by-step explanation:
(a) To find the determinant of matrix A, we can use the formula for a 2x2 matrix:
det(A) = (a*d) - (b*c)
det(A) = (5e^(3t) * (-3e^(4t))) - (2e^(4t) * (-4e^(3t)))
det(A) = -15e^(7t) + 8e^(7t) = -7e^(7t)
So, the determinant of A is -7e^(7t).
(b) To find the matrix of cofactors, we need to calculate the cofactor of each element in the matrix. The cofactor of an element a_ij is given by the formula:
C_ij = (-1)^(i+j) * det(M_ij)
Where M_ij is the submatrix obtained by removing the i-th row and j-th column from A. By applying this formula to each element, we get the following matrix of cofactors:
C = [(-3e^(4t)) (2e^(4t)); (4e^(3t)) (5e^(3t))]