matrices problem

Question: The matrix M is given by M

= \begin{pmatrix} a & b\\ c & d \end{pmatrix}

, where

a,b,c,d \epsilon \mathbb{R}

.
a)Find

M^{2}

.
b)Given that

M^{2} = M

and that b and c are non-zero, prove that M is singular.
c)Prove that in this case, the transformation T, is defined by

T:\begin{pmatrix} x\\ y \end{pmatrix} \mapsto M \begin{pmatrix} x\\ y \end{pmatrix}

maps all points of the plane to points of the line

(1-a) x =by

my attempt:

M^2 = \begin{pmatrix} a^2 + bc & ab+bd \\ ac+dc & bc+d^2 \end{pmatrix}

prove that M is singular

\begin{pmatrix} a^2 + bc & ab+bd \\ ac+dc & bc+d^2 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

if matrix is singular then

det(M) = 0

thus we need to prove that

det(M) = 0

det(M) = ad -bc

a^2+bc=a......1 \\ ab+bd = b .....2\\ ac+dc=c ......3 \\ bc+d^2=d .....4

from 1

bc = a- a^2

from 2

ab+ bd =b

bd = b - ab

bd = b(1-a)

d = 1-a

sub bc and d into

ad-bc

thus

a(1-a) - (a - a^2) = 0

since

ad -bc =0

therefore M is singular

I need help with part c
please help

(edited 5 years ago)

bump

Couple of ways, but use the fact that the second column of M contains much of the info, given that you've already found a relationship between a and d. You can also use the det expression to get the equivalent relationship for c and consider the column space of the matrix.

Original post by bigmansouf

bump

(edited 5 years ago)