Find the Inverse of the Resulting Matrix

Problem

[[8,93,0.01],[44,575,5],[1,0.1,0.1]]*[[8,24,5],[8,9,6],[0.1,1,2]]

Solution

1. Multiply the two matrices to find the resulting matrix A

A=[[8*(8)+93*(8)+0.01*(0.1),8*(24)+93*(9)+0.01*(1),8*(5)+93*(6)+0.01*(2)],[44*(8)+575*(8)+5*(0.1),44*(24)+575*(9)+5*(1),44*(5)+575*(6)+5*(2)],[1*(8)+0.1*(8)+0.1*(0.1),1*(24)+0.1*(9)+0.1*(1),1*(5)+0.1*(6)+0.1*(2)]]

2. Simplify the arithmetic for each entry of matrix A

A=[[808.001,1029.01,598.02],[4952.5,6236,3680],[8.81,25,5.8]]

3. Calculate the determinant of matrix A denoted as det(A)

det(A)=808.001*(6236⋅5.8−3680⋅25)−1029.01*(4952.5⋅5.8−3680⋅8.81)+598.02*(4952.5⋅25−6236⋅8.81)

4. Evaluate the determinant value.

det(A)=808.001*(−55831.2)−1029.01*(−3696.3)+598.02*(68873.34)

det(A)=−45111623.7912+3803529.5973+41187634.7988

det(A)=−459.3951

5. Find the adjugate matrix adj(A) by calculating the transpose of the matrix of cofactors.

(C_11)=(6236)*(5.8)−(3680)*(25)=−55831.2

(C_12)=−((4952.5)*(5.8)−(3680)*(8.81))=3696.3

(C_13)=(4952.5)*(25)−(6236)*(8.81)=68873.34

(C_21)=−((1029.01)*(5.8)−(598.02)*(25))=8982.242

(C_22)=(808.001)*(5.8)−(598.02)*(8.81)=−582.1504

(C_23)=−((808.001)*(25)−(1029.01)*(8.81))=−11134.4469

(C_31)=(1029.01)*(3680)−(598.02)*(6236)=57123.934

(C_32)=−((808.001)*(3680)−(598.02)*(4952.5))=−1355.854

(C_33)=(808.001)*(6236)−(1029.01)*(4952.5)=−56054.4085

6. Apply the formula for the inverse matrix A(−1)=1/det(A)*adj(A)

A(−1)=1/(−459.3951)*[[−55831.2,8982.242,57123.934],[3696.3,−582.1504,−1355.854],[68873.34,−11134.4469,−56054.4085]]

Final Answer

([[8,93,0.01],[44,575,5],[1,0.1,0.1]]*[[8,24,5],[8,9,6],[0.1,1,2]])(−1)≈[[121.532,−19.552,−124.346],[−8.046,1.267,2.951],[−149.922,24.237,122.018]]

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