Determin if x is an eigenvector of (5I-A). What is the corresponding eigenvalue
Solution
?=5 ? is an eigenvalue of \"-A ? Av = ?v for some v ? 0 (definition of eigenvalue) ?
(-A + ?I)x = 0 has x = v as a nontrivial solution ? -A + ?I is not invertible (invertibility criterion
for square matrices) ? det(-A + ?I) = 0. Step 1: Solve the characteristic equation det(-A + ?I) = 0
and get the eigenvalues ?1, ?2, ... . Step 2: For each eigenvalue ?i, solve the homogeneuous
system (-A + ?iI)x = 0 and get the eigenvectors with ?i as the eigenvalue. In the second step, the
answer is actually presented as a basis of nul(-A +?iI), called the eigenspace of A with
eigenvalue ?i.
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Determinant Properties of Matrices Determinant properties (b) Consid.pdf
1. Determinant Properties of Matrices Determinant properties (b) Consider a 4x4 matrix A. Add
o.6xrow 1 to row 2 to get B. Solve for BI. (c) In general la IAI for real a. Does liA ilAl also hold
for complex matrix? (d) Let B aA)n where a is a constant. Derive B
Solution
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