Question #192278

Consider the matrices


A = 3 0 2 , B = -5 1 1 , C = 1 1 1

4 -6 3 0 3 0 2 3 -1

-2 1 8 7 6 2 3 -5 -7


Verify the following expressions (where possible and give reasons)

(i) A + (B + C) = (A + B) + C and A(BC) = (AB)C.

(ii) (a − b)C = aC + bC and a(B − C) = aB − aC, where a = −2, b = 3 .

(iii) (AT)T = A and (A − B)T = A T− BT



1
Expert's answer
2021-05-17T18:40:02-0400

(i)


A+(B+C)=(302463218)A+(B+C)=\begin{pmatrix} 3 & 0 & 2 \\ 4 & -6 & 3 \\ -2 & 1 & 8 \end{pmatrix}

+((511030762)+(111231357))+\bigg(\begin{pmatrix} -5 & 1 & 1 \\ 0 & 3 & 0 \\ 7 & 6 & 2 \end{pmatrix}+\begin{pmatrix} 1 & 1 & 1\\ 2 & 3 & -1 \\ 3 & -5 & -7 \end{pmatrix}\bigg)




=(302463218)+(4222611015)=\begin{pmatrix} 3 & 0 & 2 \\ 4 & -6 & 3 \\ -2 & 1 & 8 \end{pmatrix}+\begin{pmatrix} -4 & 2 & 2 \\ 2 & 6 & -1 \\ 10 & 1 & -5 \end{pmatrix}

=(124602823)=\begin{pmatrix} -1 & 2 & 4 \\ 6 & 0 & 2 \\ 8 & 2 & 3 \end{pmatrix}

(A+B)+C=((302463218)(A+B)+C=\bigg(\begin{pmatrix} 3 & 0 & 2 \\ 4 & -6 & 3 \\ -2 & 1 & 8 \end{pmatrix}


+(511030762))+(111231357)+\begin{pmatrix} -5 & 1 & 1 \\ 0 & 3 & 0 \\ 7 & 6 & 2 \end{pmatrix}\bigg)+\begin{pmatrix} 1 & 1 & 1\\ 2 & 3 & -1 \\ 3 & -5 & -7 \end{pmatrix}

=(2134335710)+(111231357)=\begin{pmatrix} -2 & 1 & 3 \\ 4 & -3 & 3 \\ 5 & 7 & 10 \end{pmatrix}+\begin{pmatrix} 1 & 1 & 1\\ 2 & 3 & -1 \\ 3 & -5 & -7 \end{pmatrix}

=(124602823)=\begin{pmatrix} -1 & 2 & 4 \\ 6 & 0 & 2 \\ 8 & 2 & 3 \end{pmatrix}

A+(B+C)=(124602823)=(A+B)+CA+(B+C)=\begin{pmatrix} -1 & 2 & 4 \\ 6 & 0 & 2 \\ 8 & 2 & 3 \end{pmatrix}=(A+B)+C


BC=(511030762)(111231357)BC=\begin{pmatrix} -5 & 1 & 1 \\ 0 & 3 & 0 \\ 7 & 6 & 2 \end{pmatrix}\begin{pmatrix} 1 & 1 & 1\\ 2 & 3 & -1 \\ 3 & -5 & -7 \end{pmatrix}


=(0713693251513)=\begin{pmatrix} 0 & -7 & -13 \\ 6 & 9 & -3 \\ 25 & 15 & -13 \end{pmatrix}

A(BC)=(302463218)(0713693251513)A\big(BC\big)=\begin{pmatrix} 3 & 0 & 2 \\ 4 & -6 & 3 \\ -2 & 1 & 8 \end{pmatrix}\begin{pmatrix} 0 & -7 & -13 \\ 6 & 9 & -3 \\ 25 & 15 & -13 \end{pmatrix}

=(5096539377320614381)=\begin{pmatrix} 50 & 9 & -65 \\ 39 & -37 & -73 \\ 206 & 143 &-81 \end{pmatrix}

AB=(302463218)(511030762)AB=\begin{pmatrix} 3 & 0 & 2 \\ 4 & -6 & 3 \\ -2 & 1 & 8 \end{pmatrix}\begin{pmatrix} -5 & 1 & 1 \\ 0 & 3 & 0 \\ 7 & 6& 2 \end{pmatrix}

=(11571410664914)=\begin{pmatrix} -1 & 15 & 7 \\ 1 & 4 &10 \\ 66 & 49 & 14 \end{pmatrix}

(AB)C=(11571410664914)(111231357)\big(AB\big)C=\begin{pmatrix} -1 & 15& 7 \\ 1 & 4 & 10 \\ 66 & 49 & 14 \end{pmatrix}\begin{pmatrix} 1 & 1 & 1\\ 2 & 3 & -1 \\ 3 & -5 & -7 \end{pmatrix}

=(5096539377320614381)=\begin{pmatrix} 50 & 9 & -65\\ 39 & -37 & -73 \\ 206 & 143 & -81 \end{pmatrix}

A(BC)=(5096539377320614381)=A(BC)A\big(BC\big)=\begin{pmatrix} 50 & 9 & -65\\ 39 & -37 & -73 \\ 206 & 143 & -81 \end{pmatrix}=A\big(BC\big)

(ii)


a=2,b=3a=-2, b=3

(ab)C=(23)(111231357)(a-b)C=(-2-3)\begin{pmatrix} 1 & 1 & 1\\ 2 & 3 & -1 \\ 3 & -5 & -7 \end{pmatrix}

=5(111231357)=(55510155152535)=-5\begin{pmatrix} 1 & 1 & 1\\ 2 & 3 & -1 \\ 3 & -5 & -7 \end{pmatrix}=\begin{pmatrix} -5 & -5 & -5\\ -10 & -15 & 5 \\ -15 & 25 & 35\end{pmatrix}

aC=2(111231357)=(22246261014)aC=-2\begin{pmatrix} 1 & 1 & 1\\ 2 & 3 & -1 \\ 3 & -5 & -7 \end{pmatrix}=\begin{pmatrix} -2 & -2 & -2\\ -4 & -6 & 2 \\ -6 & 10 & 14 \end{pmatrix}

bC=3(111231357)=(33369391521)bC=3\begin{pmatrix} 1 & 1 & 1\\ 2 & 3 & -1 \\ 3 & -5 & -7 \end{pmatrix}=\begin{pmatrix} 3 & 3 & 3\\ 6 & 9 & -3 \\ 9 & -15 & -21 \end{pmatrix}

aCbC=(22246261014)(33369391521)aC-bC=\begin{pmatrix} -2 & -2 & -2\\ -4 & -6 & 2 \\ -6 & 10 & 14 \end{pmatrix}-\begin{pmatrix} 3 & 3 & 3\\ 6 & 9 & -3 \\ 9 & -15 & -21 \end{pmatrix}

=(55510155152535)=\begin{pmatrix} -5 & -5 & -5\\ -10 & -15 & 5 \\ -15 & 25 & 35\end{pmatrix}

(ab)C=(55510155152535)=aCbC(a-b)C=\begin{pmatrix} -5 & -5 & -5\\ -10 & -15 & 5 \\ -15 & 25 & 35\end{pmatrix}=aC-bC


BC=(511030762)(111231357)B-C=\begin{pmatrix} -5 & 1 & 1 \\ 0 & 3 & 0 \\ 7 & 6 & 2 \end{pmatrix}-\begin{pmatrix} 1 & 1 & 1\\ 2 & 3 & -1 \\ 3 & -5 & -7 \end{pmatrix}

=(6002014119)=\begin{pmatrix} -6 & 0 & 0\\ -2 & 0 & 1 \\ 4 & 11& 9 \end{pmatrix}

a(BC)=2(6002014119)=(120040282218)a(B-C)=-2\begin{pmatrix} -6 & 0 & 0\\ -2 & 0 & 1 \\ 4 & 11& 9 \end{pmatrix}=\begin{pmatrix} 12 & 0 & 0\\ 4 & 0 & -2 \\ -8 & -22 & -18 \end{pmatrix}

aB=2(511030762)=(102206014124)aB=-2\begin{pmatrix} -5 & 1 & 1 \\ 0 & 3 & 0 \\ 7 & 6 & 2 \end{pmatrix}=\begin{pmatrix} 10 & -2 & -2 \\ 0 & -6 & 0 \\ -14 & -12 &-4 \end{pmatrix}

aBaC=(102206014124)aB-aC=\begin{pmatrix} 10 & -2 & -2 \\ 0 & -6 & 0 \\ -14 & -12 &-4 \end{pmatrix}

(22246261014)=(120040282218)-\begin{pmatrix} -2 & -2 & -2\\ -4 & -6 & 2 \\ -6 & 10 & 14 \end{pmatrix}=\begin{pmatrix} 12 & 0 & 0\\ 4 & 0 & -2 \\ -8 & -22& -18 \end{pmatrix}

a(BC)=(120040282218)=aBaCa(B-C)=\begin{pmatrix} 12 & 0 & 0\\ 4 & 0 & -2 \\ -8 & -22& -18 \end{pmatrix}=aB-aC

(iii)


AT=(302463218)T=(342061238)A^T=\begin{pmatrix} 3 & 0 & 2 \\ 4 & -6 & 3 \\ -2 & 1 & 8 \end{pmatrix}^T=\begin{pmatrix} 3 & 4 & -2 \\ 0 & -6 & 1 \\ 2 & 3 & 8 \end{pmatrix}

(AT)T=(342061238)T=(302463218)=A(A^T)^T=\begin{pmatrix} 3 & 4 & -2 \\ 0 & -6 & 1 \\ 2 & 3 & 8 \end{pmatrix}^T=\begin{pmatrix} 3 & 0 & 2 \\ 4 & -6 & 3 \\ -2 & 1 & 8 \end{pmatrix}=A

(AT)T=(302463218)=A(A^T)^T=\begin{pmatrix} 3 & 0 & 2 \\ 4 & -6 & 3 \\ -2 & 1 & 8 \end{pmatrix}=A


AB=(302463218)(511030762)A-B=\begin{pmatrix} 3 & 0 & 2 \\ 4 & -6 & 3 \\ -2 & 1 & 8 \end{pmatrix}-\begin{pmatrix} -5 & 1 & 1 \\ 0 & 3 & 0 \\ 7 & 6 & 2 \end{pmatrix}

=(811493956)=\begin{pmatrix} 8 & -1 & 1 \\ 4 & -9 & 3 \\ -9 & -5 & 6 \end{pmatrix}

(AB)T=(811493956)T=(849195136)(A-B)^T=\begin{pmatrix} 8 & -1 & 1 \\ 4 & -9 & 3 \\ -9 & -5 & 6 \end{pmatrix}^T=\begin{pmatrix} 8 & 4 & -9 \\ -1 & -9 & -5 \\ 1 & 3 & 6 \end{pmatrix}

BT=(511030762)T=(507136102)B^T=\begin{pmatrix} -5 & 1 & 1 \\ 0 & 3 & 0 \\ 7 & 6 & 2 \end{pmatrix}^T=\begin{pmatrix} -5 & 0 & 7 \\ 1 & 3 & 6 \\ 1 & 0& 2 \end{pmatrix}

ATBT=(342061238)(507136102)A^T-B^T=\begin{pmatrix} 3 & 4 & -2 \\ 0 & -6 & 1 \\ 2 & 3 & 8 \end{pmatrix}-\begin{pmatrix} -5 & 0 & 7 \\ 1 & 3 & 6 \\ 1 & 0& 2 \end{pmatrix}

=(849195136)=\begin{pmatrix} 8 & 4 & -9 \\ -1 & -9 & -5 \\ 1 & 3 &6 \end{pmatrix}

(AB)T=(849195136)=ATBT(A-B)^T=\begin{pmatrix} 8 & 4 & -9 \\ -1 & -9 & -5 \\ 1 & 3 & 6 \end{pmatrix}=A^T-B^T




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