Simplify the Matrix [[-53,22,7],[52,-8,-68],[-23,-38,37]]^2

Problem

[[−53,22,7],[52,−8,−68],[−23,−38,37]]2

Solution

1. Identify the operation as matrix multiplication of the matrix A by itself, where A2=A⋅A

A=[[−53,22,7],[52,−8,−68],[−23,−38,37]]

2. Calculate the first row of the resulting matrix by taking the dot product of the first row of A with each column of A

(c_11)=(−53)*(−53)+(22)*(52)+(7)*(−23)=2809+1144−161=3792

(c_12)=(−53)*(22)+(22)*(−8)+(7)*(−38)=−1166−176−266=−1608

(c_13)=(−53)*(7)+(22)*(−68)+(7)*(37)=−371−1496+259=−1608

3. Calculate the second row of the resulting matrix by taking the dot product of the second row of A with each column of A

(c_21)=(52)*(−53)+(−8)*(52)+(−68)*(−23)=−2756−416+1564=−1608

(c_22)=(52)*(22)+(−8)*(−8)+(−68)*(−38)=1144+64+2584=3792

(c_23)=(52)*(7)+(−8)*(−68)+(−68)*(37)=364+544−2516=−1608

4. Calculate the third row of the resulting matrix by taking the dot product of the third row of A with each column of A

(c_31)=(−23)*(−53)+(−38)*(52)+(37)*(−23)=1219−1976−851=−1608

(c_32)=(−23)*(22)+(−38)*(−8)+(37)*(−38)=−506+304−1406=−1608

(c_33)=(−23)*(7)+(−38)*(−68)+(37)*(37)=−161+2584+1369=3792

Final Answer

[[−53,22,7],[52,−8,−68],[−23,−38,37]]2=[[3792,−1608,−1608],[−1608,3792,−1608],[−1608,−1608,3792]]

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