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We have learned about elementary operations

Let’s learn how to find inverse of a matrix using it.

We will find inverse of a 2 × 2 & a 3 × 3 matrix

 

Note:- While doing elementary operations, we use

Only rows

    OR

Only columns

Not both

 

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Let's take some examples
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  1. Chapter 3 Class 12 Matrices
  2. Concept wise

Transcript

Thus, We can use either ๐‘…_1โ†” ๐‘…_3 ๐‘…_1โ†’ ๐‘…_2 + ๐‘…_1 ๐‘…_3โ†’ 2๐‘…_3 OR ๐ถ_1โ†’" " ๐ถ_2 + 9๐ถ_2 ๐ถ_1โ†’ 5/2 ๐ถ_1 ๐ถ_2โ†’ ๐ถ_3โˆ’3๐ถ_1 But not ๐‘…_1โ†” ๐‘…_3 ๐‘…_1โ†’ ๐‘…_2 + ใ€–2๐‘…ใ€—_1 ๐‘ช_๐Ÿโ†’ ๐‘ช_๐Ÿ‘ + ใ€–๐Ÿ‘๐‘ชใ€—_๐Ÿ This is wrong Find inverse of [โ– 8(๐Ÿ‘&๐Ÿ@๐Ÿ&๐Ÿ’)] Let A = [โ– 8(3&2@1&4)] We know that A = A I This becomes Aโˆ’1 Convert to I using elementary transformation A = A I [โ– 8(3&2@1&4)] = A [โ– 8(1&0@0&1)] Making 3 to 1 ๐‘…_1 โ†’๐‘…_1โˆ’ 2๐‘…_2 [โ– 8(3โˆ’2 (1) &2โˆ’2(4)@1&4)] = A [โ– 8(1โˆ’2 (0) &0โˆ’2(1)@0&1)] [โ– 8(3โˆ’2 &2โˆ’8@1&4)] = A [โ– 8(1&โˆ’2@0&1)] [โ– 8(1&โˆ’6@1&4)] = A [โ– 8(1&โˆ’2@0&1)] Making 1 to 0 ๐‘…_2 โ†’๐‘…_2โˆ’ ๐‘…_1 [โ– 8(1&โˆ’6@1โˆ’1&4โˆ’(โˆ’6))] = A [โ– 8(1 &โˆ’2@0โˆ’1&1โˆ’(โˆ’2))] [โ– 8(1&โˆ’6@0&4+6)] = A [โ– 8(1&โˆ’2@โˆ’1&1+2)] [โ– 8(1&โˆ’6@0&10)] = A [โ– 8(1&โˆ’2@โˆ’1&3)] Making 10 to 1 ๐‘…_2 โ†’๐‘…_2/10 [โ– 8(1&โˆ’6@0/10&10/10)] = A [โ– 8(1 &โˆ’2@(โˆ’1)/10&3/10)] [โ– 8(1&โˆ’6@0&1)] = A [โ– 8(1 &โˆ’2@(โˆ’1)/10&3/10)] Making โˆ’6 to 0 ๐‘…_1 โ†’๐‘…_1+ 6๐‘…_2 [โ– 8(1+6(0)&โˆ’6+6(1)@0&1)] = A [โ– 8(1+6((โˆ’1)/10) &โˆ’2+6(3/10)@(โˆ’1)/10&3/10)] [โ– 8(1&โˆ’6+6@0&1)] = A [โ– 8(1โˆ’6/10&โˆ’2+18/10@(โˆ’1)/10&3/10)] [โ– 8(1&0@0&1)] = A [โ– 8(4/10&(โˆ’2)/10@(โˆ’1)/10&3/10)] This is similar to I = AAโˆ’1 Therefore, Aโˆ’1 = [โ– 8(4/10&(โˆ’2)/10@(โˆ’1)/10&3/10)] Find inverse of [โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)] Let A = [โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)] We know that A = AI [โ– 8(9&2&1@5&โˆ’1&6@4&0&โˆ’2)] = A [โ– 8(1&0&0@0&1&0@0&0&1)] Making 9 to 1 ๐‘…_1 โ†’๐‘…_1โˆ’ 2๐‘…_3 [โ– 8(9โˆ’2(4)&2โˆ’2(0)&1โˆ’2(โˆ’2)@5&โˆ’1&6@4&0&โˆ’2)] = A [โ– 8(1โˆ’2(0)&0โˆ’2(0)&0โˆ’2(1)@0&1&0@0&0&1)] [โ– 8(9โˆ’8&2โˆ’0&1+4@5&โˆ’1&6@4&0&โˆ’2)] = A [โ– 8(1&0&โˆ’2@0&1&0@0&0&1)] [โ– 8(1&2&5@5&โˆ’1&6@4&0&โˆ’2)] = A [โ– 8(1&0&โˆ’2@0&1&0@0&0&1)] Making 5 to 0 ๐‘…_2 โ†’๐‘…_2โˆ’ 5๐‘…_1 [โ– 8(1&2&5@5โˆ’5(1)&โˆ’1โˆ’5(2)&6โˆ’5(5)@4&0&โˆ’2)] = A [โ– 8(1&0&โˆ’2@0โˆ’5(1)&1โˆ’5(0)&0โˆ’5(โˆ’2)@0&0&1)] [โ– 8(1&2&5@5โˆ’5&โˆ’1โˆ’10&6โˆ’25@4&0&โˆ’2)] = A [โ– 8(1&0&โˆ’2@โˆ’5&1&10@0&0&1)] [โ– 8(1&2&5@0&โˆ’11&โˆ’19@4&0&โˆ’2)] = A [โ– 8(1&0&โˆ’2@โˆ’5&1&10@0&0&1)] Making 4 to 0 ๐‘…_3 โ†’๐‘…_3โˆ’ 4๐‘…_1 [โ– 8(1&2&5@0&โˆ’11&โˆ’19@4โˆ’4(1)&0โˆ’4(2)&โˆ’2โˆ’4(5))] = A [โ– 8(1&0&โˆ’2@โˆ’5&1&10@0โˆ’4(1)&0โˆ’4(0)&1โˆ’4(โˆ’2))] [โ– 8(1&2&5@0&โˆ’11&โˆ’19@4โˆ’4&โˆ’8&โˆ’2โˆ’20)] = A [โ– 8(1&0&โˆ’2@โˆ’5&1&10@โˆ’4&0&1+8)] [โ– 8(1&2&5@0&โˆ’๐Ÿ๐Ÿ&โˆ’19@4&โˆ’8&โˆ’22)] = A [โ– 8(1&0&โˆ’2@โˆ’5&1&10@โˆ’4&0&9)] Making โˆ’11 to 1 ๐‘…_2 โ†’๐‘…_2/(โˆ’11) [โ– 8(1&2&5@0/(โˆ’11)&(โˆ’11)/(โˆ’11)&(โˆ’19)/(โˆ’11)@0&โˆ’8&โˆ’22)] = A [โ– 8(1&0&โˆ’2@(โˆ’5)/(โˆ’11)&1/(โˆ’11)&10/(โˆ’11)@โˆ’4&0&9)] [โ– 8(1&๐Ÿ&5@0&1&19/11@0&โˆ’8&โˆ’22)] = A [โ– 8(1&0&โˆ’2@5/11&(โˆ’1)/11&(โˆ’10)/11@โˆ’4&0&9)] Making 2 as 0 ๐‘…_1 โ†’๐‘…_1โˆ’ 2๐‘…_2 [โ– 8(1โˆ’2(0)&2โˆ’2(1)&5โˆ’2(19/11)@0&1&19/11@0&โˆ’8&โˆ’22)] = A [โ– 8(1โˆ’2(5/11)&0โˆ’2((โˆ’1)/11)&โˆ’2โˆ’2((โˆ’10)/11)@5/11&(โˆ’1)/11&(โˆ’10)/11@โˆ’4&0&9)] [โ– 8(1&0&5โˆ’38/11@0&1&19/11@0&โˆ’8&โˆ’22)] = A [โ– 8(1โˆ’10/11&2/11&โˆ’2+20/11@5/11&(โˆ’1)/11&(โˆ’10)/11@โˆ’4&0&9)] [โ– 8(1&0&17/11@0&1&19/11@0&โˆ’๐Ÿ–&โˆ’22)] = A [โ– 8(1/11&2/11&(โˆ’2)/11@5/11&(โˆ’1)/11&(โˆ’10)/11@โˆ’4&0&9)] Making โˆ’8 as 0 ๐‘…_3 โ†’๐‘…_3 + 8๐‘…_2 [โ– 8(1&0&17/11@0&1&19/11@0+8(0)&โˆ’8+8(1)&โˆ’22+8(19/11) )] = A [โ– 8(1/11&2/11&(โˆ’2)/11@5/11&(โˆ’1)/11&(โˆ’10)/11@โˆ’4+8(5/11)&0+8((โˆ’1)/11)&9+8((โˆ’10)/11) )] [โ– 8(1&0&17/11@0&1&19/11@0&โˆ’8+8&โˆ’22+152/11)] = A [โ– 8(1/11&2/11&(โˆ’2)/11@5/11&(โˆ’1)/11&(โˆ’10)/11@โˆ’4+40/11&(โˆ’8)/11&9โˆ’80/11)] [โ– 8(1&0&17/11@0&1&19/11@0&0&(โˆ’๐Ÿ—๐ŸŽ)/๐Ÿ๐Ÿ)] = A [โ– 8(1/11&2/11&(โˆ’2)/11@5/11&(โˆ’1)/11&(โˆ’10)/11@(โˆ’4)/11&(โˆ’8)/11&19/11)] We Make (โˆ’90)/11 to 1 ๐‘…_3 โ†’๐‘…_3 ร— (โˆ’11)/90 [โ– 8(1&0&17/11@0&1&19/11@0ร—(โˆ’11)/90&0ร—(โˆ’11)/90&(โˆ’90)/11ร—(โˆ’11)/90)] = A [โ– 8(1/11&2/11&(โˆ’2)/11@5/11&(โˆ’1)/11&(โˆ’10)/11@โˆ’4ร—(โˆ’11)/90&(โˆ’8)/11ร—(โˆ’11)/90&19/11ร—(โˆ’11)/90)] [โ– 8(1&0&17/11@0&1&19/11@0&0&1)] = A [โ– 8(1/11&2/11&(โˆ’2)/11@5/11&(โˆ’1)/11&(โˆ’10)/11@2/45&4/45&(โˆ’19)/90)] We Make 17/11 to 0 ๐‘…_1 โ†’๐‘…_1 โ€“ 17/11 ๐‘…_3 [โ– 8(1โˆ’17/11(0)&0โˆ’17/11(0)&17/11โˆ’17/11(1)@0&1&19/11@0&0&1)] = A [โ– 8(1/11โˆ’17/11 (2/45)&2/11โˆ’17/11 (4/45)&(โˆ’2)/11โˆ’17/11 ((โˆ’19)/90)@5/11&(โˆ’1)/11&(โˆ’10)/11@2/45&4/45&(โˆ’19)/90)] [โ– 8(1โˆ’17/11(0)&0โˆ’17/11(0)&17/11โˆ’17/11(1)@0&1&19/11@0&0&1)] = A [โ– 8(1/11 (1โˆ’34/45) &2/11 (1โˆ’34/35)&1/11 (โˆ’2+323/90)@5/11&(โˆ’1)/11&(โˆ’10)/11@2/45&4/45&(โˆ’19)/90)] [โ– 8(1&0&0@0&1&19/11@0&0&1)] = A [โ– 8(1/11ร—11/45&2/11ร—11/45&1/11ร—143/90@5/11&(โˆ’1)/11&(โˆ’10)/11@2/45&4/45&(โˆ’19)/90)] [โ– 8(1&0&0@0&1&19/11@0&0&1)] = A [โ– 8(1/45&2/45&13/90@5/11&(โˆ’1)/11&(โˆ’10)/11@2/45&4/45&(โˆ’19)/90)] We Make 19/11 as 1 ๐‘…_2 โ†’๐‘…_2 โˆ’ 19/11 ๐‘…_3 [โ– 8(1&0&0@0โˆ’19/11(0)&1โˆ’19/11(0)&19/11โˆ’19/11(1)@0&0&1)] = A [โ– 8(1/45&2/11&13/90@5/11โˆ’19/11 (2/45)&(โˆ’1)/11โˆ’19/11 (4/45)&(โˆ’10)/11โˆ’19/11 ((โˆ’19)/11)@2/45&4/45&(โˆ’19)/90)] [โ– 8(1&0&0@0&1&0@0&0&1)] = A [โ– 8(1/45&2/45&13/90@1/11 (5โˆ’38/45)&(โˆ’1)/11ร—121/45&1/11ร—((โˆ’539)/90)@2/45&4/45&(โˆ’19)/90)] [โ– 8(1&0&0@0&1&0@0&0&1)] = A [โ– 8(1/45&2/45&13/90@17/45&(โˆ’11)/45&(โˆ’49)/90@2/45&4/45&(โˆ’19)/90)] I = AAโˆ’1 Therefore, Aโˆ’1 = [โ– 8(1/45&2/45&13/90@17/45&(โˆ’11)/45&(โˆ’49)/90@2/45&4/45&(โˆ’19)/90)]

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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.