Let `A=[0-tan(alpha//2)tan(alpha//2)0]` and `I` be the identity matrix of order 2. Show that `I+A=(I-A)[cosalpha-sinalphasinalphacosalpha]` .

Question

Let `A=[0-tan(alpha//2)tan(alpha//2)0]` and `I` be the identity matrix of order 2. Show that `I+A=(I-A)[cosalpha-sinalphasinalphacosalpha]` .

Let `A=[0-tan(alpha//2)tan(alpha//2)0]` and `I` be the identity matrix of order 2. Show that `I+A=(I-A)[cosalpha-sinalphasinalphacosalpha]` .
A. `-I+A`
B. `I-A`
C. `-I-A`
D. none of these

Monjur Alahi

5 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - B
Since `I=[(1,0),(0,1)]` and given `A=[(0,tan alpha//2),(-tan alpha//2,0)]`
`:. I-A=[(1,-tan alpha//2),(tan alpha//2,1)]`
Now, `(I-A) [(cos alpha,-sin alpha),(sin alpha,cos alpha)]`
`=[(1,tan alpha//2),(-tan alpha//2,1)][(cos alpha,-sin alpha),(sin alpha,cos alpha)]`
`=[(1,tan alpha//2),(-tan alpha//2,1)][((1-tan^(2) alpha//2)/(1+tan^(2)alpha//2),-(2 tan alpha//2)/(1+tan^(2) alpha//2)),((2 tan alpha//2)/(1+tan^(2) alpha//2),(1-tan^(2) alpha//2)/(1+tan^(2) alpha//2))]`
`=[((1-tan^(2) alpha//2)/(1+tan^(2)alpha//2)+(2 tan^(2) alpha//2)/(1+tan^(2) alpha//2),-(2 tan alpha//2)/(1+tan^(2) alpha//2)),((-tan alpha//2 (1-tan^(2) alpha//2))/(1+tan^(2) alpha//2)+(2 tan alpha//2)/(1+tan^(2) alpha//2),(2 tan^(2) alpha//2)/(1+tan^(2) alpha//2)+(1-tan^(2) alpha//2)/(1+tan^(2) alpha//2))]`
`=[(((1+tan^(2) alpha//2))/((1+tan^(2) alpha//2)),-(tan alpha//2(1+tan^(2) alpha//2))/((1+tan^(2) alpha//2))),((tan alpha//2(1+tan^(2) alpha//2))/((1+tan^(2) alpha//2)),((1+tan^(2) alpha//2))/((1+tan^(2) alpha//2)))]`
`=[(1,-tan alpha//2),(tan alpha//2,a)]`
`=I-A` [Using (1)]