Question

1. P =
`\begin{pmatrix}-\sin \left(x\right)&\cos \left(x\right)\\ \cos \left(x\right)&\sin \left(x\right)\end{pmatrix}`

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

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

Answer 1

We know that:

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

#1

The values:

• 
  `P = \left[\begin{array}{cc}-sinx&cosx\\cosx&sinx\\\end{array}\right] = -sin^2x-cos^2x= -1`

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

• 
  `I = \left[\begin{array}{cc}1&0\\0&1\\\end{array}\right] = 1-0=1`

The sum:

• PQ + 2I = (-1)*1 + 2*1 = -1 + 2 = 1

#2

The values:

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

The sum:

• PQ + 2I = 1*1 + 2*1 = 1 + 2 = 3