Question

0 1 (c) Consider the matrix0 2 -2 k 3 i. Compute the determinant. ii. For what value(s) of k does A exist? iii. For what value(s) of k does the linear system A7 = 7 have nontrivial solutions? iv. For what value(s) of k does A have zero as an eigenvalue? v. For any vector be R*, find the value(s) of k for which the linear system A7 = b has a unique solution.

Answer 1

[co] Consider the Matrix K 1 0 2. K K -2 (1) Compute the determinant. Any: K 1 TAL ОК 2 -2 K 3 2 о к TAI KIK 2 K 3 O 11 ox -2 3 TAL K (3x-2x) -0 + 1 (0+2K) K²+2k [1] For What value(s) of k Alexisto Amy A-l exist iff determinant of A is not equal to Zero. Alto K272k to K(K + 2) to k to or kt - 2 hence for all her except 0 and 2 A-" exist. hence

[1] For What valus of k does the linear system Ano have montrival solutions? Amy :-) : An=0 is called homogeneous system. If A-t exist or we say determinant is not zero then this system have atrival Solution which is Zero Solution, But when determinant is zero then this system have mon- trival solution. det (A)=0 K? + 2K = 0 K(+2) = 0 K=o or k=-2 this system have non trival =) salution, [iv] For What values of K does A have Zero as eigen value ? Ang Matoix A have Zero eigen value iff it's determinant is zero : A have three possible eigen value de, de, da And we to know that dede da = det (A) hence when det(A) is Zero then conly A have Zero eigen value. =) K=o or k=-2 A have zero eigen value.

[vi] for any vector BER? find the values) of k for which the linear system An= 5 has a unique Solution. Any:n then If system Rank A) Añ=5 have a unique solution Rank ([A16]) = Number of Variable u ER3 our number of variable 3 =) Rank (A) should be 3. that is if Rank (A) = 3 it is only possible When det (A) to hence K(K+2) to K to, kf 2 a for all value of Kell except K=0 and K=2 system An= to have Unique solution