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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 â‹¯ a 1p â‹¯ a 1q â‹¯ a 1n a 21 a 22 â‹¯ a 2p â‹¯ a 2q â‹¯ a 2n â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a i1 a i2 â‹¯ a ip â‹¯ a iq â‹¯ a in â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a j1 a j2 â‹¯ a jp â‹¯ a jq â‹¯ a jn â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a m1 a m2 â‹¯ a mp â‹¯ a mq â‹¯ a mn

Interchange of rows or columns

= a 11 a 12 â‹¯ a 1p â‹¯ a 1q â‹¯ a 1n a 21 a 22 â‹¯ a 2p â‹¯ a 2q â‹¯ a 2n â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a i1 a i2 â‹¯ a ip â‹¯ a iq â‹¯ a in â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a j1 a j2 â‹¯ a jp â‹¯ a jq â‹¯ a jn â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a m1 a m2 â‹¯ a mp â‹¯ a mq â‹¯ a mn     R i â†’ R j â†’         a 11 a 12 â‹¯ a 1p â‹¯ a 1q â‹¯ a 1n a 21 a 22 â‹¯ a 2p â‹¯ a 2q â‹¯ a 2n â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a j1 a j2 â‹¯ a jp â‹¯ a jq â‹¯ a jn â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a i1 a i2 â‹¯ a ip â‹¯ a iq â‹¯ a in â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a m1 a m2 â‹¯ a mp â‹¯ a mq â‹¯ a mn

Ex:

2 -1 4 5 2 3 -2 7 1        R 2 â†’ R 3 â†’         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 â‹¯ a 1p â‹¯ a 1q â‹¯ a 1n a 21 a 22 â‹¯ a 2p â‹¯ a 2q â‹¯ a 2n â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a i1 a i2 â‹¯ a ip â‹¯ a iq â‹¯ a in â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a j1 a j2 â‹¯ a jp â‹¯ a jq â‹¯ a jn â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a m1 a m2 â‹¯ a mp â‹¯ a mq â‹¯ a mn     C p â†’ C q â†’   a 11 a 12 â‹¯ a 1q â‹¯ a 1p â‹¯ a 1n a 21 a 22 â‹¯ a 2q â‹¯ a 2p â‹¯ a 2n â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a i1 a i2 â‹¯ a iq â‹¯ a ip â‹¯ a in â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a j1 a j2 â‹¯ a jq â‹¯ a jp â‹¯ a jn â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a m1 a m2 â‹¯ a mq â‹¯ a mp â‹¯ 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 â†’ C 1 â†’       -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 â†’ kRi, k â‰  0

= a 11 a 12 â‹¯ a 1p â‹¯ a 1q â‹¯ a 1n a 21 a 22 â‹¯ a 2p â‹¯ a 2q â‹¯ a 2n â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a i1 a i2 â‹¯ a ip â‹¯ a iq â‹¯ a in â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a j1 a j2 â‹¯ a jp â‹¯ a jq â‹¯ a jn â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a m1 a m2 â‹¯ a mp â‹¯ a mq â‹¯ a mn     R i â†’ kR i â†’     a 11 a 12 â‹¯ a 1p â‹¯ a 1q â‹¯ a 1n a 21 a 22 â‹¯ a 2p â‹¯ a 2q â‹¯ a 2n â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® ka i1 ka i2 â‹¯ ka ip â‹¯ ka iq â‹¯ ka in â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a j1 a j2 â‹¯ a jp â‹¯ a jq â‹¯ a jn â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a m1 a m2 â‹¯ a mp â‹¯ a mq â‹¯ a mn

Ex:

2 -1 4 5 2 3 -2 7 1     R 2 â†’ 1/5R 2 â†’     2 -1 4 1 2/5 3/5 -2 7 1    Second row is multiplied with 1/5

Cp â†’ lCp, l â‰  0

= a 11 a 12 â‹¯ a 1p â‹¯ a 1q â‹¯ a 1n a 21 a 22 â‹¯ a 2p â‹¯ a 2q â‹¯ a 2n â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a i1 a i2 â‹¯ a ip â‹¯ a iq â‹¯ a in â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a j1 a j2 â‹¯ a jp â‹¯ a jq â‹¯ a jn â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a m1 a m2 â‹¯ a mp â‹¯ a mq â‹¯ a mn     C p â†’ lC p â†’     a 11 a 12 â‹¯ la 1p â‹¯ a 1q â‹¯ a 1n a 21 a 22 â‹¯ la 2p â‹¯ a 2q â‹¯ a 2n â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a i1 a i2 â‹¯ la ip â‹¯ a iq â‹¯ a in â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a j1 a j2 â‹¯ la jp â‹¯ a jq â‹¯ a jn â‹® â‹® â‹® â‹® â‹® â‹® â‹® â‹® a m1 a m2 â‹¯ la mp â‹¯ a mq â‹¯ a mn

Ex:

2 -1 4 5 2 3 -2 7 1     C 2 â†’ 4C 2 â†’     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 â†’ Ri + kRj, k â‰  0

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

Ex:   2 -1 4 5 2 3 -2 7 1   R 2 â†’ R 2 + 3 R 3 â†’     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 â†’ Cp + lCq, l â‰  0

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

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

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