Question #262380

Let A =

(8 -1 2

2 0 -5)


B =

(-1 7

3 -2

1 5)


and C =

(2 1

3 5)


(a) Calculate AB and A + B if they exist.

(b) Verify that (AB)C = A(BC).

(c) Calculate C-1 A.


1
Expert's answer
2021-11-09T16:02:03-0500

a)

(A+B) does not exist, since matrices are of different sizes (2x3 and 3x2)

AB=(812205)(173215)=(83+256+2+10251425)=(968711)AB=\begin{pmatrix} 8 & -1 &2\\ 2 & 0&-5 \end{pmatrix}\begin{pmatrix} -1 & 7 \\ 3 & -2\\ 1&5 \end{pmatrix}=\begin{pmatrix} -8-3+2 & 56+2+10 \\ -2-5 & 14-25 \end{pmatrix}=\begin{pmatrix} -9 & 68 \\ -7 & -11 \end{pmatrix}


b)

(AB)C=(968711)(2135)=(18+2049+3401433755)=(1863314762)(AB)C=\begin{pmatrix} -9 & 68 \\ -7 & -11 \end{pmatrix}\begin{pmatrix} 2 & 1 \\ 3 & 5 \end{pmatrix}=\begin{pmatrix} -18+204 & -9+340 \\ -14-33 & -7-55 \end{pmatrix}=\begin{pmatrix} 186 & 331 \\ -47 & -62 \end{pmatrix}


BC=(173215)(2135)=(2+211+35663102+151+25)=(1934071726)BC=\begin{pmatrix} -1 & 7 \\ 3 & -2\\ 1&5 \end{pmatrix}\begin{pmatrix} 2 & 1 \\ 3 & 5 \end{pmatrix}=\begin{pmatrix} -2+21 & -1+35 \\ 6-6 & 3-10\\ 2+15&1+25 \end{pmatrix}=\begin{pmatrix} 19 & 34 \\ 0 & -7\\ 17&26 \end{pmatrix}


A(BC)=(812205)(1934071726)=(152+34272+7+52388568130)=A(BC)=\begin{pmatrix} 8 & -1 &2\\ 2 & 0&-5 \end{pmatrix}\begin{pmatrix} 19 & 34 \\ 0 & -7\\ 17&26 \end{pmatrix}=\begin{pmatrix} 152+34 & 272+7+52 \\ 38-85 & 68-130 \end{pmatrix}=


=(1863314762)=\begin{pmatrix} 186 & 331 \\ -47 & -62 \end{pmatrix}


c)

for C-1:

C=(5312)C_*=\begin{pmatrix} 5 & -3 \\ -1 & 2 \end{pmatrix} , CT=(5132)C_*^T=\begin{pmatrix} 5 & -1 \\ -3 & 2 \end{pmatrix}


C=103=7|C|=10-3=7


C1=CTC==17(5132)C^{-1}=\frac{C_*^T}{|C|}=\frac{}{}=\frac{1}{7}\begin{pmatrix} 5 & -1 \\ -3 & 2 \end{pmatrix}


C1A=17(5132)(812205)=17(402510+524+43610)=C^{-1}A=\frac{1}{7}\begin{pmatrix} 5 & -1 \\ -3 & 2 \end{pmatrix}\begin{pmatrix} 8 & -1 &2\\ 2 & 0&-5 \end{pmatrix}=\frac{1}{7}\begin{pmatrix} 40-2 & -5 &10+5\\ -24+4 & 3&-6-10 \end{pmatrix}=


=17(3851520316)=\frac{1}{7}\begin{pmatrix} 38 & -5 &15\\ -20 & 3&-16 \end{pmatrix}


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