Matrices

The system can be written in matrix form , where: The solution is .
Step 1: Find the determinant of A. Step 2: Find the adjugate of A.
The matrix of cofactors is: The adjugate is the transpose of the cofactor matrix: Step 3: Find the inverse of A. Step 4: Calculate . Thus, .

Step 1: Recall the properties of the transpose of a matrix. One of the fundamental properties is that for any scalar and any matrix , the transpose of their product is the scalar multiplied by the transpose of the matrix.
Step 2: Applying this property, we have . The statement is therefore correct.

Step 1: Multiply the first two matrices. Step 2: Multiply the resulting matrix by the column vector. Step 3: Set the resulting matrix equal to the right-hand side and solve the system of equations. From the second row: . From the first row: . Substitute : The problem does not yield a unique solution for and . The solution is and any that satisfy .