Find the Inverse 1/-5*[[19,4],[48,-7]]

Problem

−1/5*[[19,4],[48,−7]]

Solution

1. Identify the matrix A and the scalar k Let A=[[19,4],[48,−7]] and k=−1/5 The inverse of a scalar multiple of a matrix is given by (k*A)(−1)=1/k*A(−1)

2. Calculate the determinant of A For a 2×2 matrix [[a,b],[c,d]] the determinant is det(A)=a*d−b*c

det(A)=(19)*(−7)−(4)*(48)

det(A)=−133−192

det(A)=−325

3. Find the inverse of matrix A The formula is A(−1)=1/det(A)*[[d,−b],[−c,a]]

A(−1)=1/(−325)*[[−7,−4],[−48,19]]

4. Apply the scalar property (k*A)(−1)=1/k*A(−1) Here, 1/k=−5

(k*A)(−1)=−5⋅1/(−325)*[[−7,−4],[−48,19]]

5. Simplify the scalar multiplication.

(k*A)(−1)=(−5)/(−325)*[[−7,−4],[−48,19]]

(k*A)(−1)=1/65*[[−7,−4],[−48,19]]

Final Answer

(−1/5*[[19,4],[48,−7]])(−1)=[[−7/65,−4/65],[−48/65,19/65]]

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