Question

If A = $\left[\begin{array}{ccc}0&tan(\alpha/2)\\tan(\alpha/2)&0\\\end{array}\right]$ and I is a 2×2 unit matrix then (I-A) $\left[\begin{array}{ccc}cos\alpha&-sin\alpha\\sin\alpha&sin\alpha\\\end{array}\right]$ is
(a) -I+A
(b) I-A
(c) -I-A
(d) I+A

Answer 1

Given :  A =  $\begin{bmatrix} 0 &- tan(\frac{\alpha}{2}) \\ tan(\frac{\alpha}{2}) & 0 \end{bmatrix}$      I is a 2×2 unit matrix

To find :   (I - A) $\begin{bmatrix} cos\alpha &- sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}$

Solution:

I is a 2×2 unit matrix

=>  I = $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

I  - A   =  $\begin{bmatrix} 1 & tan(\frac{\alpha}{2}) \\ -tan(\frac{\alpha}{2}) & 1 \end{bmatrix}$

(I  - A)  $\begin{bmatrix} cos\alpha &- sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}$

=  $\begin{bmatrix} 1 & tan(\frac{\alpha}{2}) \\ -tan(\frac{\alpha}{2}) & 1 \end{bmatrix} \begin{bmatrix} cos\alpha &- sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}$

= $\begin{bmatrix} cos\alpha + tan(\frac{\alpha}{2}) sin\alpha & -sin\alpha + tan(\frac{\alpha}{2}) cos\alpha \\ -tan(\frac{\alpha}{2}) cos\alpha + sin\alpha & +tan(\frac{\alpha}{2}) sin\alpha + cos\alpha \end{bmatrix}$

cosα  = cos²(α/2) - sin²(α/2)

sinα = 2sin(α/2)cos(α/2)

cosα  +tan(α/2) sinα  =  cos²(α/2) - sin²(α/2) + +tan(α/2)2sin(α/2)cos(α/2)

= cos²(α/2)  - sin²(α/2)  +  2sin²(α/2)

= cos²(α/2)  + sin²(α/2)

= 1

-sinα  +tan(α/2) cosα  = -2sin(α/2)cos(α/2) + tan(α/2) (cos²(α/2) - sin²(α/2))

=  -2sin(α/2)cos(α/2) + sin(α/2)cos(α/2)  - tan(α/2) sin²(α/2)

= -sin(α/2)cos(α/2)  - tan(α/2) sin²(α/2)

= -tan(α/2)(cos²(α/2)) - tan(α/2) sin²(α/2)

=  -tan(α/2)( cos²(α/2) + sin²(α/2))

=  -tan(α/2)

Similarly :    -tan(α/2) cosα + sinα = - (-sinα  +tan(α/2) cosα) = - ( -tan(α/2)) = tan(α/2)

=   $\begin{bmatrix} 1 &- tan(\frac{\alpha}{2}) \\ tan(\frac{\alpha}{2}) & 1 \end{bmatrix}$

=  $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$  +  $\begin{bmatrix} 0 &- tan(\frac{\alpha}{2}) \\ tan(\frac{\alpha}{2}) & 0 \end{bmatrix}$

= I  + A

(I - A) $\begin{bmatrix} cos\alpha &- sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}$ =  I + A  

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निम्नलिखित ज्ञात कीजिए   (i) A + B

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I is a 2×2 unit matrix then (I-A)

https://brainly.in/question/16490188

Answer 2

Answer:

-I+A is the answer of

If A = $\left[\begin{array}{ccc}0&tan(\alpha/2)\\tan(\alpha/2)&0\\\end{array}\right]$ and I is a 2×2 unit matrix then (I-A) $\left[\begin{array}{ccc}cos\alpha&-sin\alpha\\sin\alpha&sin\alpha\\\end{array}\right]$ is