Notes On Elementary Operations on a Matrix - CBSE Class 12 Maths

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     

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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