# Ex 4.5, 11 - Chapter 4 Class 12 Determinants

Last updated at Jan. 22, 2020 by Teachoo

Last updated at Jan. 22, 2020 by Teachoo

Transcript

Ex 4.5, 11 Find the inverse of each of the matrices [■8(1&0&0@0&cos𝛼&sin𝛼@0&sin𝛼&−cos𝛼 )] Let A =[■8(1&0&0@0&cos𝛼&sin𝛼@0&sin𝛼&−cos𝛼 )] We know that A–1 = 1/(|A|) (adj A) exists if |A|≠ 0 Calculating |A| |A| = |■8(1&0&0@0&cos𝛼&sin𝛼@0&sin𝛼&−cos𝛼 )| = 1 |■8(cos𝛼&sin𝛼@sin𝛼&−cos𝛼 )|– 0 |■8(0&sin𝛼@0&〖− cos〗𝛼 )|+ 0|■8(0&cos𝛼@0&𝑠𝑖𝑛 𝛼)| = 1(– cos2α – sin2α ) – 0 + 0 = –( cos2α + sin2α ) = –1 Since |𝐴|≠ 0 Thus A-1 exists Calculating adj A adj (A) = [■8(A11&A21&A31@A12&A22&A32@A13&A23&A33)] A = [■8(1&0&0@0&cos𝛼&sin𝛼@0&sin𝛼&−cos𝛼 )] M11 = |■8(cos"α" &sin"α" @sin"α" &−cos"α" )| = –cos2α – sin2α = –(cos^2α 〖+ 𝑠𝑖𝑛〗^2α ) = –1 M12 = |■8(0&sin 𝛼@0&−cos 𝛼)| = 0 – 0 = 0 M13 = |■8(0&cos𝛼@0&sin 𝛼)| = 0 – 0 = 0 M21 = |■8(0&0@sin 𝛼&−cos𝛼 )| = 0 – 0 = 0 M22 = |■8(1&0@0&−cos 𝛼)| = –cos α – 0 = –cos α M23 = |■8(1&0@0&sin𝛼 )| = sin α = 0 = sin α M31 = |■8(0&0@cos 𝛼&sin 𝛼)| = 0 – 0 = –0 M32 = |■8(1&0@0&sin 𝛼)| = sin α – 0 = sin α M33 = |■8(1&0@0&cos 𝛼)| = cos α + 0 = cos α Now, A11 = 〖(−1)〗^(1+1) M11 = 〖(−1)〗^2 (–1)2 = –1 A12 = 〖(−1)〗^(1+2) M12 = 〖(−1)〗^3 0 = 0 A13 = 〖(−1)〗^(1+3) M13 = 〖(−1)〗^4 0 = 0 A21 = 〖(−1)〗^(2+1)M21 = (–1)3 0 = 0 A22 = 〖(−1)〗^(2+2) M22 = 〖(−1)〗^4(– cos α) = –cos α A23 = 〖"(– 1)" 〗^(2+3) M23 = 〖"(–1)" 〗^5 sin α = –sin α A31 = 〖(−1)〗^(3+1). M31 = 〖(−1)〗^4 0 = 0 A32 = 〖(−1)〗^(3+2)sin α = (–1)5 sin α = –sin α A33 = 〖(−1)〗^(3+3)M33 = (–1)6 cos α = cos α So, adj (A) = [■8(A11&A21&A31@A12&A22&A32@A33&A23&A33)] = [■8(−1&0&0@0&−cos𝛼&−sin𝛼@0&−sin𝛼&cos𝛼 )] Calculating inverse Now, A– 1 = 1/(|A|) ( adj (A)) = 1/(−1) [■8(−1&0&0@0&−cos𝛼&−sin𝛼@0&−sin𝛼&cos𝛼 )] = – [■8(−1&0&0@0&−cos𝛼&−sin𝛼@0&−sin𝛼&cos𝛼 )] = [■8(𝟏&𝟎&𝟎@𝟎&𝒄𝒐𝒔𝜶&𝒔𝒊𝒏𝜶@𝟎&𝒔𝒊𝒏𝜶&〖−𝒄𝒐𝒔〗𝜶 )]

Chapter 4 Class 12 Determinants

Concept wise

- Finding determinant of a 2x2 matrix
- Evalute determinant of a 3x3 matrix
- Area of triangle
- Equation of line using determinant
- Finding Minors and cofactors
- Evaluating determinant using minor and co-factor
- Find adjoint of a matrix
- Finding Inverse of a matrix
- Inverse of two matrices and verifying properties
- Finding inverse when Equation of matrice given
- Checking consistency of equations
- Find solution of equations- Equations given
- Find solution of equations- Statement given
- Verifying properties of a determinant
- Two rows or columns same
- Whole row/column zero
- Whole row/column one
- Making whole row/column one and simplifying
- Proving Determinant 1 = Determinant 2
- Solving by simplifying det.
- Using Property 5 (Determinant as sum of two or more determinants)

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.