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

Let `A=[{:(1,-2,3),(-4,2,5):}]" and "B=[{:(2,3),(4,5),(-2,1):}].` Find AB and BA, and show that `ABneBA.`

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Here A is a `2xx3` matrix and B is a `3xx2` matrix.
So, AB exists and it is a `2xx2` matrix.
Now, `AB=[{:(1,-2,3),(-4," "2,5):}][{:(2,3),(4,5),(-2,1):}]`
`=[{:(1.2+(-2).4+3.(-2),1.3+(-2).5+3.1),((-4).2+2.4+5.(-2),(-4).3+2.5+5.1):}]`
`=[{:(-12,-4),(-10," "3):}].`
Again, B is a `3xx2` matrix and A is a `2xx3` matrix.
So, BA exists and it is a `3xx3` matrix.
Now, `BA=[{:(2,3),(4,5),(-2,1):}][{:(1,-2,3),(-4," "2,5):}]`
`=[{:(2.1+3.(-4),2.(-2)+3.2,2.3+3.5),(4.1+5.(-4),4.(-2)+5.2,4.3+5.5),((-2).1+1.(-4),(-2).(-2)+1.2,(-2).3+1.5):}]`
`=[{:(-10,2,21),(-16,2,37),(-6,6,-1):}].`
Hence, `ABneBA.`

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RS AGGARWAL-MATRICES-Exercise 5F

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