matrices problem Watch

bigmansouf
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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
Last edited by bigmansouf; 3 weeks ago
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bigmansouf
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mqb2766
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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)
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Last edited by mqb2766; 3 weeks ago
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