Let , where…

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Let

A(k)=\begin{pmatrix}2 & k \\ 3 & 6\end{pmatrix}

, where

k\in\mathbb R

. (i) State the criterion for the invertibility of a square matrix in terms of its determinant. (ii) Determine the value of

k

for which

A(k)

is not invertible. (iii) For the homogeneous system

A(k)\,X=0

, where

X=\begin{pmatrix}x\\y\end{pmatrix}

, explain what happens to the solution set when

A(k)

is singular and when it is non-singular.