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Elementary Operations on a Matrix

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Elementary Operations on a Matrix - Lesson Summary

Row and column operations on a matrix.

These operations are particularly helpful when finding the inverse of a matrix, solving a system of linear equations and finding determinants.

There are six operations available on a matrix.

1. Three on rows

2. Three on columns

We have a matrix, A, with m rows and n columns.

A = a 11 a 12 ⋯ a 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ a ip ⋯ a iq ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp ⋯ a mq ⋯ a mn

Interchange of rows or columns

= a 11 a 12 ⋯ a 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ a ip ⋯ a iq ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp ⋯ a mq ⋯ a mn     R i → R j →         a 11 a 12 ⋯ a 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ a ip ⋯ a iq ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp ⋯ a mq ⋯ a mn       

Ex:

         2 -1 4 5 2 3 -2 7 1        R 2 → R 3 →         2 -1 4 -2 7 1 5 2 3       We have interchanged rows R2 and R3

Now, interchange the columns of a matrix.

= a 11 a 12 ⋯ a 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ a ip ⋯ a iq ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp ⋯ a mq ⋯ a mn     C p → C q →   a 11 a 12 ⋯ a 1q ⋯ a 1p ⋯ a 1n a 21 a 22 ⋯ a 2q ⋯ a 2p ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ a iq ⋯ a ip ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jq ⋯ a jp ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mq ⋯ a mp ⋯ a mn      
              

Here, we have interchanged the elements of the columns Cp and Cq.

Ex:

   2 -1 4 5 2 3 -2 7 1     C 2 → C 1 →       -1 2 4 2 5 3 7 -2 1       We interchange the first and second columns.

Multiplication of a row or column by a non-zero number.

Ri → kRi, k ≠ 0

= a 11 a 12 ⋯ a 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ a ip ⋯ a iq ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp ⋯ a mq ⋯ a mn     R i → kR i →     a 11 a 12 ⋯ a 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ka i1 ka i2 ⋯ ka ip ⋯ ka iq ⋯ ka in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp ⋯ a mq ⋯ a mn

Ex:  

  2 -1 4 5 2 3 -2 7 1     R 2 → 1/5R 2 →     2 -1 4 1 2/5 3/5 -2 7 1    Second row is multiplied with 1/5

Cp → lCp, l ≠ 0

= a 11 a 12 ⋯ a 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ a ip ⋯ a iq ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp ⋯ a mq ⋯ a mn     C p → lC p →     a 11 a 12 ⋯ la 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ la 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ la ip ⋯ a iq ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ la jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ la mp ⋯ a mq ⋯ a mn


Ex: 

  2 -1 4 5 2 3 -2 7 1     C 2 → 4C 2 →     2 -4 4 5 8 3 -2 28 1         Second column is multiplied by 4.

Addition of elements of a row or column to the corresponding elements of any other row or column multiplied by a non-zero number

Ri → Ri + kRj, k ≠ 0

A = a 11 a 12 ⋯ a 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ a ip ⋯ a iq ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp ⋯ a mq ⋯ a mn

First, we will multiply the jth row with the non-zero constant, k.

Then we add these elements to the corresponding elements in the ith row


A = a 11 a 12 ⋯ a 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 + k × a j1 a i2 +k × a j2 ⋯ a ip +k × a jp ⋯ a iq +k × a jq ⋯ a in +k × a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp ⋯ a mq ⋯ a mn


Ex:   2 -1 4 5 2 3 -2 7 1   R 2 → R 2 + 3 R 3 →     2 -1 4 5+3x(-2) 2+3(7) 3+3(1) -2 7 1    2 -1 4 -1 23 6 -2 7 1                       

A similar type of operation over the columns of a matrix is

Cp → Cp + lCq, l ≠ 0


A =a 11 a 12 ⋯ a 1p ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ a ip ⋯ a iq ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp ⋯ a mq ⋯ a mn


A =  a 11 a 12 ⋯ a 1p +l × a 1q ⋯ a 1q ⋯ a 1n a 21 a 22 ⋯ a 2p +l × a 2q ⋯ a 2q ⋯ a 2n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a i1 a i2 ⋯ a ip +l × a iq ⋯ a iq ⋯ a in ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a j1 a j2 ⋯ a jp +l × a jq ⋯ a jq ⋯ a jn ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a m1 a m2 ⋯ a mp +l × a mq ⋯ a mq ⋯ a mn

Ex:  2 -1 4 5 2 3 -2 7 1    C 1 → C 1 + 2 C 3 →       2+2x4 -1 4 5+2x3 2 3 -2+2x1 7 1   =    10 -1 4 11 2 3 0 7 1     

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