Find the Inverse [[d,j,n],[g,k,o],[h,l,p]]=7

Problem

[[d,j,n],[g,k,o],[h,l,p]](−1)

Solution

1. Identify the matrix and its determinant. The problem provides a 3×3 matrix and states that its determinant is 7 Let A=[[d,j,n],[g,k,o],[h,l,p]] and det(A)=7

2. Recall the formula for the inverse of a matrix. The inverse is the adjugate matrix divided by the determinant.

A(−1)=1/det(A)*adj(A)

3. Calculate the matrix of cofactors. Each element (C_i*j) is the determinant of the 2×2 submatrix remaining after removing row i and column j multiplied by (−1)(i+j)

(C_11)=k*p−o*l

(C_12)=−(g*p−o*h)

(C_13)=g*l−k*h

(C_21)=−(j*p−n*l)

(C_22)=d(p)−n*h

(C_23)=−(d(l)−j*h)

(C_31)=j*o−n*k

(C_32)=−(d(o)−n*g)

(C_33)=d(k)−j*g

4. Transpose the cofactor matrix to find the adjugate matrix adj(A)

adj(A)=[[k*p−o*l,−(j*p−n*l),j*o−n*k],[−(g*p−o*h),d(p)−n*h,−(d(o)−n*g)],[g*l−k*h,−(d(l)−j*h),d(k)−j*g]]

5. Divide the adjugate matrix by the determinant, which is 7

A(−1)=1/7*[[k*p−o*l,n*l−j*p,j*o−n*k],[o*h−g*p,d(p)−n*h,n*g−d(o)],[g*l−k*h,j*h−d(l),d(k)−j*g]]

Final Answer

[[d,j,n],[g,k,o],[h,l,p]](−1)=[[(k*p−o*l)/7,(n*l−j*p)/7,(j*o−n*k)/7],[(o*h−g*p)/7,(d(p)−n*h)/7,(n*g−d(o))/7],[(g*l−k*h)/7,(j*h−d(l))/7,(d(k)−j*g)/7]]

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