ZJSG09112007
contestada

1. P = [tex]\begin{pmatrix}-\sin \left(x\right)&\cos \left(x\right)\\ \cos \left(x\right)&\sin \left(x\right)\end{pmatrix}[/tex]
Q = [tex]\begin{pmatrix}\sin \left(x\right)&\cos \left(x\right)\\ \cos \left(x\right)&-\sin \left(x\right)\end{pmatrix}[/tex]
and I is a 2×2 identity matrix. Find PQ +2I

2. P = [tex]\begin{pmatrix}\cos \left(x\right)&-\sin \left(x\right)\\ \sin \left(x\right)&\cos \left(x\right)\end{pmatrix}[/tex]
Q =[tex]\begin{pmatrix}\cos \left(x\right)&\sin \left(x\right)\\ -\sin \left(x\right)&\cos \left(x\right)\end{pmatrix}[/tex]
and I is a 2×2 identity matrix. Find PQ +2I

Respuesta :

mhanifa

Answer:

We know that:

  • [tex]\left[\begin{array}{cc}a&b\\c&d\end{array}\right] = ad - bc[/tex]
  • The 2x2 identity matrix has a = d = 1 and b = c = 0
  • sin²x + cos²x = 1

#1

The values:

  • [tex]P = \left[\begin{array}{cc}-sinx&cosx\\cosx&sinx\\\end{array}\right] = -sin^2x-cos^2x= -1[/tex]
  • [tex]Q = \left[\begin{array}{cc}sinx&cosx\\cosx&-sinx\\\end{array}\right] = sin^2x-(-cos^2x)= sin^2x+cos^2x=1[/tex]
  • [tex]I = \left[\begin{array}{cc}1&0\\0&1\\\end{array}\right] = 1-0=1[/tex]

The sum:

  • PQ + 2I = (-1)*1 + 2*1 = -1 + 2 = 1

#2

The values:

  • [tex]P = \left[\begin{array}{cc}cosx&-sinx\\sinx&cosx\\\end{array}\right] = cos^2x-(-sin^2x)= 1[/tex]
  • [tex]Q = \left[\begin{array}{cc}cosx&sinx\\-sinx&cosx\\\end{array}\right] = cos^2x-(-sin^2x)= 1[/tex]
  • [tex]I = \left[\begin{array}{cc}1&0\\0&1\\\end{array}\right] = 1-0=1[/tex]

The sum:

  • PQ + 2I = 1*1 + 2*1 = 1 + 2 = 3
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