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Subsection 1.3 Algebraic properties of matrix operations

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Subsubsection 1.3.1 Properties of matrix sum

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For the properties shown below, Am×n,Bm×n, and Cm×n are matrices and r and s are scalars.
  • Commutative property.
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    The sum of two matrices is commutative:
    Am×n+Bm×n=Bm×n+Am×n
    A=[431−10−2],B=[20−135−1].
    A+B=[431−10−2]+[20−135−1]=[63025−3].
    B+A=[20−135−1]+[431−10−2]=[63025−3].
  • Associative property.
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    Matrix addition is associative :
    Am×n+(Bm×n+Cm×n)=(Am×n+Bm×n)+Cm×n.
    If we have more than two matrices to add, we can first sum two of them and then add the third matrix to this result. This property tells us that the order in which we choose the first two matrices does not matter.
    Consider the following three matrices
    A=[2103],B=[−1012],C=[0−41−2].
    • Note that A+B+C can be obtained by first adding A and B and then adding the resulting matrix to C:
      A+B+C=(A+B)+C.
      (A+B)=[2103]+[−1012]=[1115]
      (A+B)+C=[1115]+[0−41−2]=[1−323].
    • Alternatively, A+B+C can be obtained by first adding B and C and then adding the resulting matrix to A:
      A+B+C=A+(B+C).
      (B+C)=[−1012]+[0−41−2]=[−1−420]
      A+(B+C)=[2103]+[−1−420]=[1−323].
    • Another alternative is to find A+B+C by first adding A and C and then adding the resulting matrix to B:
      A+B+C=(A+C)+B.
      (A+C)=[2103]+[0−41−2]=[2−311]
      (A+C)+B=[2−311]+[−1012]=[1−323].
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Subsubsection 1.3.2 Properties of matrix-scalar multiplication

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For the properties shown below, Am×n,Bm×n, and Cm×n are matrices and r and s are scalars.
  • Distributive property 1.
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    Multiplication of a matrix by a scalar is distributive with respect to matrix addition:
    r(Am×n+Bm×n)=rAm×n+rBm×n.
    4×([−120413−21]+[3−1112−4−22])=4×[−120413−21]+4×[3−1112−4−22]
    4×([21153−1−43])=[−48016412−84]+[12−4448−16−88][8442012−4−1612]=[8442012−4−1612]
  • Distributive property 2.
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    Multiplication of a matrix by a scalar is distributive with respect to the scalar addition:
    (r+s)Cm×n=rCm×n+sCm×n.
    (2+4)×[−2101]=2×[−2101]+4×[−2101]6×[−2101]=[−4202]+[−8404][−12606]=[−12606]
  • Associative property.
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    Multiplication of a matrix by a scalar is associative with respect to the scalar multiplication:
    r(s×Cm×n)=(rs)Cm×n.
    −1×(2×[1−1320−1300])=(−1×2)×[1−1320−1300]−1×[2−2640−2600]=−2×[1−1320−1300][−22−6−402−600]=[−22−6−402−600]
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Subsubsection 1.3.3 Properties of matrix-matrix multiplication

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For the properties shown below, A,B, and C are matrices and r is a scalar. The sizes for the matrices will be given for each property.
  • Associative property.
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    Matrix-matrix multiplication is associative:
    A(BC)=(AB)CAm×n(Bn×pCp×q)=(Am×nBn×p)Cp×q.
    A3×2(B2×4C4×3)=(A3×2B2×4)C4×3[11−2031]([01−132−11−2][1−1−32314100−21])=([11−2031][01−132−11−2])[1−1−32314100−21]
    [11−2031][−6−5440−9]=[20−110−24−622−57][1−1−32314100−21][−2−5−51210−8−14−153]3×3=[−2−5−51210−8−14−153]3×3
  • Distributive property.
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    Matrix-matrix multiplication is distributive with respect to matrix addition:
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      Case 1:
      A(B+C)=AB+ACAm×n(Bn×p+Cn×p)=Am×nBn×p+Am×nCn×p
      A3×2(B2×4+C2×4)=A3×2B2×4+A3×2C2×4[11−2031]([01−132−11−2]+[1−1−312310])=[11−2031][01−132−11−2]+[11−2031][1−1−312310]
      [11−2031][10−44422−2]=[20010−22−622−27]+[32−21−226−250−83][52−22−208−872−1010]=[52−22−208−872−1010]
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      Case 2:
      (A+B)C=AC+BC(Am×n+Bm×n)Cn×p=Am×nCn×p+Bm×nCn×p
      (A2×2+B2×2)C2×3=A2×2C2×3+B2×2C2×3([1001]+[3−214])[21013−1]=[1001][21013−1]+[3−214][21013−1]
      [4−215][21013−1]=[21013−1]+[4−32613−4][6−22716−5]=[6−22716−5]
  • Associative property with scalar multiplication.
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    Matrix multiplication is associative with respect to scalar multiplication:
    r(AB)=(rA)B=A(rB)r(Am×nBn×p)=(rA)m×nBn×p=Am×n(rB)n×p
    2(A2×2B2×2)=(2A)2×2B2×2=A2×2(2B)2×22([3210][1−121])=(2[3210])[1−121]=[3210](2[1−121])2[7−11−1]=[6430][1−121]=[3210][2−242]
    [14−22−2]=[14−22−2]=[14−22−2]
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