Matrix multiplication is not commutative
i.e. `AB != BA` (in general)
e.g `A =[(1,1),(0,0)]; B = [(1,0),(0,0)]; AB= [(1,0),(0,0)]` and `BA= [(1,1),(0,0)]`
In fact if `AB` is defined it is possible that `AB` is not defined or may have different order
`tt(( , A , B),
((1), 3 xx 2 , 2 xx 3),
((2), 2 xx 2 , 2 xx 3))`
then `AB` is of order `3xx3 ` and `BA` is of order `2 xx 2`
Note :
(i) If `AB= 0 ⇏` that one of the matrices is zero however if any one of either `A` or `B` is null matrix then
`AB = 0` provided the product exist.
(ii) If `AB = AC ⇏ B = C` but if `B = C => AB = AC`
(iii) In case `AB = BA` is restrict of matrices `A` and `B` the two matrices are said to commute each other one if
`AB =- BA` then they are said to anticommute each other
e.g (i) `A =[(a,0),(0,b)]` and `B= [(c,0),(0,b)] \ \ \ \ \ \ [AB =BA]`
Note that multiplication of diagonal matrices of the same order will be commutative.
(iv) For every square matrix `A`, there exist an identity matrix of the same order such that
`IA = AI = A` where `I` is the unit matrix of the same order.
(v) If `A = 0` then `det. A = 0`, however if `det. A = 0 ⇏ A = 0`
(vi) Martix Multiplication is Associative:
If `A, B` & `C` are conformable for the product `AB` & `BC` , then
`(A. B) . C = A. (B . C)`
`A = [a_(ij)]` is `m xx n ; B = [b_(ij)]` is `n xx p ; C = [c_(ij) ]` is `p x q`
(vii) Distributivity :
`tt((A(B+C)=AB+AC),((A+B)C = AC + BC))]` Provided `A, B` & `C` are conformable for respective products
`A = m xx n ;B = n xx p ; c = n xx p`
(viii) If `I` be unit matrix, then `I^2 = I^3 = ... = I^m = I (m in I)`
(viii) If A and Bare two matrices of the same order, then :
`=> (A+ B)^2 =A^2 + AB + BA + B^2`
`=>(A- B)^2 =A^2 - AB - BA+ B^2 `
`=> (A -B)(A +B)= A^2 + AB-BA+ B^2 `
`=> (A+ B)(A -B)= A^2 -AB+ BA-B^2`
`=>A(-B) = (-A)(B) = -AB `
Matrix multiplication is not commutative
i.e. `AB != BA` (in general)
e.g `A =[(1,1),(0,0)]; B = [(1,0),(0,0)]; AB= [(1,0),(0,0)]` and `BA= [(1,1),(0,0)]`
In fact if `AB` is defined it is possible that `AB` is not defined or may have different order
`tt(( , A , B),
((1), 3 xx 2 , 2 xx 3),
((2), 2 xx 2 , 2 xx 3))`
then `AB` is of order `3xx3 ` and `BA` is of order `2 xx 2`
Note :
(i) If `AB= 0 ⇏` that one of the matrices is zero however if any one of either `A` or `B` is null matrix then
`AB = 0` provided the product exist.
(ii) If `AB = AC ⇏ B = C` but if `B = C => AB = AC`
(iii) In case `AB = BA` is restrict of matrices `A` and `B` the two matrices are said to commute each other one if
`AB =- BA` then they are said to anticommute each other
e.g (i) `A =[(a,0),(0,b)]` and `B= [(c,0),(0,b)] \ \ \ \ \ \ [AB =BA]`
Note that multiplication of diagonal matrices of the same order will be commutative.
(iv) For every square matrix `A`, there exist an identity matrix of the same order such that
`IA = AI = A` where `I` is the unit matrix of the same order.
(v) If `A = 0` then `det. A = 0`, however if `det. A = 0 ⇏ A = 0`
(vi) Martix Multiplication is Associative:
If `A, B` & `C` are conformable for the product `AB` & `BC` , then
`(A. B) . C = A. (B . C)`
`A = [a_(ij)]` is `m xx n ; B = [b_(ij)]` is `n xx p ; C = [c_(ij) ]` is `p x q`
(vii) Distributivity :
`tt((A(B+C)=AB+AC),((A+B)C = AC + BC))]` Provided `A, B` & `C` are conformable for respective products
`A = m xx n ;B = n xx p ; c = n xx p`
(viii) If `I` be unit matrix, then `I^2 = I^3 = ... = I^m = I (m in I)`
(viii) If A and Bare two matrices of the same order, then :
`=> (A+ B)^2 =A^2 + AB + BA + B^2`
`=>(A- B)^2 =A^2 - AB - BA+ B^2 `
`=> (A -B)(A +B)= A^2 + AB-BA+ B^2 `
`=> (A+ B)(A -B)= A^2 -AB+ BA-B^2`
`=>A(-B) = (-A)(B) = -AB `