Let A denote the 2 × 2 matrix [: ] (a) Show that the characteristic equation of A (Eq. (8), Section 5.2) is given by 12 – (a + d)A + (ad – bc) = 0. (b) Suppose that the eigenvalues of A are pure imaginary. Show that the trace T(A) = a +d of A must be zero and that the determinant D(A) = ad – bc must be positive. Conclude that c # 0.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Using 2 x 2 Matrix. solve the problem mentioned in the Attachment.

Let A denote the 2 × 2 matrix
[: ]
(a) Show that the characteristic equation of A (Eq. (8),
Section 5.2) is given by
12 – (a + d)A + (ad – bc) = 0.
(b) Suppose that the eigenvalues of A are pure imaginary.
Show that the trace T(A) = a +d of A must be zero
and that the determinant D(A) = ad – bc must be
positive. Conclude that c # 0.
Transcribed Image Text:Let A denote the 2 × 2 matrix [: ] (a) Show that the characteristic equation of A (Eq. (8), Section 5.2) is given by 12 – (a + d)A + (ad – bc) = 0. (b) Suppose that the eigenvalues of A are pure imaginary. Show that the trace T(A) = a +d of A must be zero and that the determinant D(A) = ad – bc must be positive. Conclude that c # 0.
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