When the eigenvalues of-[:]a bA =0 dare 11O and 12 = 3, what are the possible values of a and d? (Select all that apply.)O a = -3 and d = 0O a = 0 and d = 0a = 3 and d = 0D a =3 and d = 3O a =O and d3O a = 0 and d-3

A matrix is a mathematical object containing numbers arranged in rows and columns. If the number of rows of the first matrix is equal to the number of columns of the second matrix. In this problem, a square matrix ab0d is given and its eigenvalues are λ1=0 and λ2=3. We have to calculate the possible values for a and d. The size of the square matrix will be the same as the order of the characteristic equation. The characteristic equation of matrix A is defined as |A-λI|=0. The eigenvalues of the matrix are defined as the solution of the characteristic equation. To find the possible values of a,d set up the eigenvalue equation of the matrix A=ab0d. Since the matrix is a 2×2 matrix, the characteristic equation will be quadratic. a-λb0d-λ=0(a-λ)(d-λ)-b·0=0λ2-(a+d)λ+ad=0 ···(1) According to the property of the quadratic equation, if λ1, λ2 are the roots of the quadratic equation λ2+Aλ+B=0, then λ1+λ2=-A and λ1λ2=B. Applying these results in equation (1) gives λ1+λ2=a+d and λ1λ2=ad. In this problem the eigenvalues are λ1=0 and λ2=3, therefore λ1+λ2=3 or a+d=3, Similarly λ1λ2=0 or ad=0. Check which option is satisfying these conditions Case 1; a=-3, d=0: a+d=-3≠3 Case 2; a=0, d=0: a+d=0≠3 Case 3; a=3, d=0: a+d=3ad=0 Case 4; a=3, d=3: a+d=6≠3 Case 5; a=0, d=3: a+d=3ad=0 Case 6: a=0, d=-3 a+d=-3≠3 Only cases 3 and 5 agree on the conditions a+d=3 and ad=0, hence the correct options are 3 and 5.