Matrix, Determinant and System of Linear Equations
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Welcome to our exploration of matrices, determinants, and systems of linear equations. These mathematical tools are fundamental in solving complex problems across science and engineering. Today we'll solve a simple system: x plus y equals 3, and 2x minus y equals 0. We can represent this system in matrix form as A times x equals b, where A is our coefficient matrix, x is our variable vector, and b is our constant vector.
Now let's calculate the determinant of our coefficient matrix A. The determinant is crucial because it tells us whether our system has a unique solution. For a 2 by 2 matrix, the determinant formula is a times d minus b times c. Substituting our values: 1 times negative 1 minus 1 times 2, which equals negative 1 minus 2, giving us negative 3. Since the determinant is negative 3, which is not zero, our system has a unique solution.
Now we'll apply Cramer's Rule to find our solution. Cramer's Rule states that x equals the determinant of A sub x divided by the determinant of A, and y equals the determinant of A sub y divided by the determinant of A. To find A sub x, we replace the first column of matrix A with our constant vector b. This gives us the matrix with 3, 1 in the first row and 0, negative 1 in the second row. Its determinant is negative 3. For A sub y, we replace the second column with b, giving us 1, 3 in the first row and 2, 0 in the second row, with determinant negative 6.
Now we can find our solution using the determinants we calculated. For x, we divide the determinant of A sub x by the determinant of A: negative 3 divided by negative 3 equals 1. For y, we divide the determinant of A sub y by the determinant of A: negative 6 divided by negative 3 equals 2. Therefore, our solution is x equals 1 and y equals 2.
Finally, let's verify our solution by substituting x equals 1 and y equals 2 back into the original equations. For the first equation, x plus y equals 3: 1 plus 2 equals 3, which is correct. For the second equation, 2x minus y equals 0: 2 times 1 minus 2 equals 0, which is also correct. Our solution is verified! This demonstrates how matrices, determinants, and Cramer's Rule provide a systematic approach to solving systems of linear equations.