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Byju's Answer
Standard XII
Mathematics
Inverse of a Matrix
Using element...
Question

Using elementary operations, find the inverse of the following matrix: ⎛⎜⎝−112123311⎞⎟⎠

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Solution

Let A=⎡⎢⎣−112123311⎤⎥⎦
For applying elementary row operation we write,
A = I. A
⎡⎢⎣−112123311⎤⎥⎦=⎡⎢⎣100010001⎤⎥⎦A
Applying R1↔R2, we get
⎡⎢⎣123−112311⎤⎥⎦=⎡⎢⎣010100001⎤⎥⎦A
Applying R2→R2+R1 and R3→R3−3R1, we get
⎡⎢⎣1230350−5−8⎤⎥⎦=⎡⎢⎣0101100−31⎤⎥⎦A
Applying R1→R1−23R2, we get
⎡⎢ ⎢⎣10−130350−5−8⎤⎥ ⎥⎦=⎡⎢ ⎢⎣−231301100−31⎤⎥ ⎥⎦A
Applying R2→13R2, we get
⎡⎢ ⎢ ⎢⎣10−1301530−5−8⎤⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣−23130131300−31⎤⎥ ⎥ ⎥⎦A
Applying R3→R3+5R2, we get
⎡⎢ ⎢ ⎢⎣10−1301530013⎤⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢ ⎢⎣−231301313053−431⎤⎥ ⎥ ⎥ ⎥⎦A
Applying R1→R1+R3 and R2→−5R3
⎡⎢ ⎢⎣1000100013⎤⎥ ⎥⎦=⎡⎢ ⎢⎣111−87−553−431⎤⎥ ⎥⎦A
Applying R3→3R3, we get
⎡⎢⎣100010001⎤⎥⎦=⎡⎢⎣1−11−87−55−43⎤⎥⎦A
Hence A−1=⎡⎢⎣1−11−87−55−43⎤⎥⎦

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