Find the Kernel [[3,1],[4,1]]x=[[5,-1],[2,3]]

Problem

[[3,1],[4,1]]*X=[[5,−1],[2,3]]

Solution

1. Identify the problem type. The task asks to find the "Kernel" of the equation A*X=B which in this context refers to solving for the matrix X

2. Set up the matrix equation. Let A=[[3,1],[4,1]] X=[[(x_11),(x_12)],[(x_21),(x_22)]] and B=[[5,−1],[2,3]]

3. Calculate the inverse of matrix A The determinant is |A|=(3)*(1)−(1)*(4)=−1

4. Apply the formula for the inverse of a 2×2 matrix A(−1)=1/|A|*[[d,−b],[−c,a]]

A(−1)=1/(−1)*[[1,−1],[−4,3]]

A(−1)=[[−1,1],[4,−3]]

5. Solve for X by multiplying both sides by A(−1) on the left, such that X=A(−1)*B

X=[[−1,1],[4,−3]]*[[5,−1],[2,3]]

6. Perform matrix multiplication.

(x_11)=(−1)*(5)+(1)*(2)=−3

(x_12)=(−1)*(−1)+(1)*(3)=4

(x_21)=(4)*(5)+(−3)*(2)=14

(x_22)=(4)*(−1)+(−3)*(3)=−13

Final Answer

[[3,1],[4,1]]*X=[[5,−1],[2,3]]⇒X=[[−3,4],[14,−13]]

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