Two matrices are given: C= pmatrix 1 &0 &1 2 &5 &10 0 &0 &0 pmatrix , D= pmatrix 3 &6…

MATHEMATICS · Subjective · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Two

3\times3

matrices are given:

C=\begin{pmatrix}1 &0 &1\\ 2 &5 &10\\ 0 &0 &0\end{pmatrix}, \qquad D=\begin{pmatrix}3 &6 &9\\ 1 &2 &3\\ 0 &1 &2\end{pmatrix}.

(i) Without carrying out a complete determinant expansion, decide whether

C

is singular or non-singular and justify your answer. (ii) Calculate

\det(D)

and hence state whether

D

is invertible. (iii) A system

C\,\mathbf{x}=\mathbf{b}

is to be solved for

\mathbf{x}

. Comment on the existence of a unique solution in the light of your result from part (i).