Find the Inverse

Problem

1/75*[[−7,1,−5],[8,−44,−5],[17,−56,−20]]

Solution

1. Identify the matrix A and the scalar k Let A=[[−7,1,−5],[8,−44,−5],[17,−56,−20]] The given expression is M=1/75*A The inverse is M(−1)=75*A(−1)

2. Calculate the determinant of A using the first row.

det(A)=−7*((−44)*(−20)−(−5)*(−56))−1*((8)*(−20)−(−5)*(17))−5*((8)*(−56)−(−44)*(17))

det(A)=−7*(880−280)−1*(−160+85)−5*(−448+748)

det(A)=−7*(600)−1*(−75)−5*(300)

det(A)=−4200+75−1500=−5625

3. Find the matrix of cofactors C

(C_11)=(−44)*(−20)−(−5)*(−56)=600

(C_12)=−((8)*(−20)−(−5)*(17))=75

(C_13)=(8)*(−56)−(−44)*(17)=300

(C_21)=−((1)*(−20)−(−5)*(−56))=300

(C_22)=(−7)*(−20)−(−5)*(17)=225

(C_23)=−((−7)*(−56)−(1)*(17))=−375

(C_31)=(1)*(−5)−(−5)*(−44)=−225

(C_32)=−((−7)*(−5)−(−5)*(8))=−75

(C_33)=(−7)*(−44)−(1)*(8)=300

4. Form the adjugate matrix adj(A) by transposing the cofactor matrix.

adj(A)=[[600,300,−225],[75,225,−75],[300,−375,300]]

5. Compute A(−1) using the formula A(−1)=1/det(A)*adj(A)

A(−1)=1/(−5625)*[[600,300,−225],[75,225,−75],[300,−375,300]]

6. Multiply by the scalar 75 to find M(−1)

M(−1)=75⋅1/(−5625)*[[600,300,−225],[75,225,−75],[300,−375,300]]

M(−1)=−1/75*[[600,300,−225],[75,225,−75],[300,−375,300]]

7. Simplify the final matrix by dividing each element by −75

M(−1)=[[−8,−4,3],[−1,−3,1],[−4,5,−4]]

Final Answer

(1/75*[[−7,1,−5],[8,−44,−5],[17,−56,−20]])(−1)=[[−8,−4,3],[−1,−3,1],[−4,5,−4]]

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