Give an example of a 3 x 3 matrix A such that Nul(A) = {0}.

Answered: Give an example of a 3 x 3 matrix A…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Give an example of a 3 x 3 matrix A such that Nul(A) = {0}.

Expert Solution

Step 1 Finding the matrix

Let $T : ℝ^{3} \to ℝ^{3}$ be a linear transformation defined by $T (x, y, z) = (x - y, y - z, z - x)$

Therefore $K e r (T) = \{(x, y, z) \in R^{3} : T (x, y, z) = 0\}$

$∴ x - y = 0 y - z = 0 z - x = 0 ∴ x = y = z$

$∴ K e r (T) = \{(x, y, z)\} = (1, 1, 1) ∴ N u l l (T) = d i m (K e r (T)) = 1 \neq 0$

$T (x, y, z) = (x - y, y - z, z - x) T (1, 0, 0) = (1, 0, - 1) = a (1, 0, 0) + b (0, 1, 0) + c (0, 0, 1) = (a, b, c) T (0, 1, 0) = (- 1, 1, 0) = e (1, 0, 0) + f (0, 1, 0) + g (0, 0, 1) = (d, e, f) T (0, 0, 1) = (0, - 1, 1) = h (1, 0, 0) + i (0, 1, 0) + k (0, 0, 1) = (g, h, i) A (T) = [\begin{matrix} a & d & g \\ b & e & h \\ c & f & i \end{matrix}] = [\begin{matrix} 1 & - 1 & 0 \\ 0 & 1 & - 1 \\ - 1 & 0 & 1 \end{matrix}]$