Chapter D, Exercise T20, Page 366
Question
n = 2, \(\alpha\) = 2
A : \[ \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]
det(\(\alpha\) A) = det(3A)
A <- matrix(c(1,3,2,4), nrow=2, ncol=2)
alpha = 2
n = 2
alphaA <- alpha * A
det_alphaA <- det(alphaA)
alpha_n <- alpha^n
detA <- det(A)
alpha_n_detA <- alpha_n * detA\(\alpha ^ {n}\)detA = -8
det(\(\alpha\)A) = -8
Another Example
n = 3, \(\alpha\) = 5
A : \[ \begin{pmatrix} 1 & 2 & 3\\ 3 & 4 & 5\\ 4 & 5 & 6 \end{pmatrix} \]
A <- matrix(c(1,3,4,2,4,5,3,5,6), nrow=3, ncol=3)
alpha = 5
n = 3
alphaA <- alpha * A
det_alphaA <- det(alphaA)
alpha_n <- alpha^n
detA <- det(A)
alpha_n_detA <- alpha_n * detA\(\alpha ^ {n}\)detA = 0
det(\(\alpha\)A) = 0