Determine if Linear [[2,3],[1,4]]A=[[3,0],[1,2]]

Problem

[[2,3],[1,4]]*A=[[3,0],[1,2]]

Solution

1. Identify the given matrix equation as a linear system of the form X*A=B or M*A=B Here, let M=[[2,3],[1,4]] and B=[[3,0],[1,2]]

2. Calculate the determinant of matrix M to determine if it is invertible.

det(M)=(2)*(4)−(3)*(1)

det(M)=8−3=5

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

M(−1)=1/5*[[4,−3],[−1,2]]

4. Solve for A by multiplying both sides of the equation by M(−1) from the left.

A=M(−1)*B

A=1/5*[[4,−3],[−1,2]]*[[3,0],[1,2]]

5. Perform matrix multiplication to find the entries of A

A=1/5*[[(4)*(3)+(−3)*(1),(4)*(0)+(−3)*(2)],[(−1)*(3)+(2)*(1),(−1)*(0)+(2)*(2)]]

A=1/5*[[12−3,0−6],[−3+2,0+4]]

A=1/5*[[9,−6],[−1,4]]

6. Distribute the scalar to reach the final matrix form.

A=[[1.8,−1.2],[−0.2,0.8]]

Final Answer

[[2,3],[1,4]]*A=[[3,0],[1,2]]⇒A=[[1.8,−1.2],[−0.2,0.8]]

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