Simplify the Matrix [[2,3,-2],[-1,1,0],[-3,1,-2]]^2

Problem

[[2,3,−2],[−1,1,0],[−3,1,−2]]2

Solution

1. Set up the matrix multiplication by writing the matrix multiplied by itself.

A2=[[2,3,−2],[−1,1,0],[−3,1,−2]]*[[2,3,−2],[−1,1,0],[−3,1,−2]]

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

(c_11)=(2)*(2)+(3)*(−1)+(−2)*(−3)=4−3+6=7

(c_12)=(2)*(3)+(3)*(1)+(−2)*(1)=6+3−2=7

(c_13)=(2)*(−2)+(3)*(0)+(−2)*(−2)=−4+0+4=0

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

(c_21)=(−1)*(2)+(1)*(−1)+(0)*(−3)=−2−1+0=−3

(c_22)=(−1)*(3)+(1)*(1)+(0)*(1)=−3+1+0=−2

(c_23)=(−1)*(−2)+(1)*(0)+(0)*(−2)=2+0+0=2

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

(c_31)=(−3)*(2)+(1)*(−1)+(−2)*(−3)=−6−1+6=−1

(c_32)=(−3)*(3)+(1)*(1)+(−2)*(1)=−9+1−2=−10

(c_33)=(−3)*(−2)+(1)*(0)+(−2)*(−2)=6+0+4=10

Final Answer

[[2,3,−2],[−1,1,0],[−3,1,−2]]2=[[7,7,0],[−3,−2,2],[−1,−10,10]]

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