Inverse of a matrix
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Elementary row and column operations can be used to find the inverse of a matrix. These operations include interchanging rows/columns, multiplying rows/columns by a non-zero number, and adding rows/columns. To find the inverse, the identity matrix is written next to the original matrix. Operations are performed to reduce the original matrix to the identity matrix, leaving the inverse in the identity matrix columns. Examples show using row operations to reduce a 3x3 matrix to its inverse.
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Inverse of a matrix
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- 2. By an elementary row(column) operation, we mean i) Interchange of 2 rows or columns. ii) Multiplying every element of a row or column by a non zero real number. iii) Adding to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero real number. math videos www.mathews.co.in
- 3. To find the inverse of a matrix A , by elementary row operations, we write A=I A where I is the identity matrix Elementary row operations applied to A will apply to I as well Find the inverse of 120 352 011 math videos www.mathews.co.in
- 4. We find the inverse using elementary row operations A 100 010 001 120 352 011 Try to make the lower triangle zero 122 2RRR A 100 012 001 120 370 011 math videos www.mathews.co.in
- 5. 322 3RRR To get 1 as the next diagonal entry A 100 312 001 120 010 011 233 2RRR math videos www.mathews.co.in
- 6. A 724 312 001 100 010 011 211 RRR A 724 312 311 100 010 001 AI 724 312 311 math videos www.mathews.co.in
- 7. Post multiply both sides by 1 A 11 724 312 311 AAIA 724 312 311 1 A Find the inverse of 113 321 210 math videos www.mathews.co.in
- 8. We write A= AI Elementary column operations applied to A will also apply to I 100 010 001 113 321 210 A We try to make the upper triangle zero math videos www.mathews.co.in
- 9. 100 001 010 131 312 201 A 21 CC 133 2CCC 100 201 010 131 112 001 A math videos www.mathews.co.in
- 10. 233 CCC 100 201 110 231 012 001 A 33 2 1 CC 2 1 00 101 2 1 10 131 012 001 A math videos www.mathews.co.in
- 11. We now try to make the lower triangle zero 322 3CCC 2/12/30 131 2/12/10 101 012 001 A 311 CCC math videos www.mathews.co.in
- 12. 2/12/32/1 132 2/12/12/1 100 012 001 A 211 2CCC 2/12/32/5 134 2/12/12/1 100 010 001 A 2/12/32/5 134 2/12/12/1 AI math videos www.mathews.co.in
- 13. Premultiply both sides by 1 A 2/12/32/5 134 2/12/12/1 11 AAIA 2/12/32/5 134 2/12/12/1 1 A math videos www.mathews.co.in ***************************