If A=[{:(2,-1,3),(-4," "5,1):}]" and "B=...

If `A=[{:(2,-1,3),(-4," "5,1):}]" and "B=[{:(2,3),(4,-2),(1,5):}]` then find AB and BA.
Show that `ABneBA.`

Text Solution

Verified by Experts

Here, A is a `2xx3` matrix and B is a `3xx2` matrix.
So, number of columns in A =number of rows in B.
`:.` AB exists and it is a `2xx2` matrix.
`AB=[{:(2,-1,3),(-4," "5,1):}][{:(2,3),(4,-2),(1,5):}]`
`=[{:(2.2+(-1).4+3.1," "2.3+(-1).(-2)+3.5),(-4.2+5.4+1.1,-4.3+5.(-2)+1.5):}]`
`=[{:(4-4+3,6+2+15),(-8+20+1,-12-10+5):}]=[{:(3," "23),(13,-17):}].`
Again, B is a `3xx2` matrix and A is a `2xx3` matrix.
So, number of columns in B=number of rows in A.
`:.` BA exists and it is a `3xx3` matrix.
`BA=[{:(2,3),(4,-2),(1,5):}][{:(2,-1,3),(-4," "5,1):}]`
`=[{:(2.2+3.(-4),2.(-1)+3.5,2.3+3.1),(4.2+(-2).(-4),4.(-1)+(-2).5,4.3+(-2).1),(1.2+5.(-4),1.(-1)+5.5,1.3+5.1):}]`
`=[{:(4-12,-2+15,6+3),(8+8,-4-10,12-2),(2-20,-1+25,3+5):}]=[{:(-8,13,9),(16,-14,10),(-18,24,8):}].`
Clearly, `ABneBA.`

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