Step 1: Identify the matrix A. A = | 1 2 | | 3 5 |.
Step 2: Calculate the determinant of A. Use the formula det(A) = (1*5) - (2*3).
Step 3: Perform the multiplication in the determinant formula: 1*5 = 5 and 2*3 = 6.
Step 4: Subtract the results: 5 - 6 = -1. So, det(A) = -1.
Step 5: Now, we need to find det(2A). The formula for the determinant of a scaled matrix is det(kA) = k^n * det(A), where n is the size of the matrix. Here, n = 2 (since A is 2x2).
Step 6: Calculate k^n. Here, k = 2, so 2^2 = 4.
Step 7: Now, multiply this result by det(A): det(2A) = 4 * det(A).
Step 8: Substitute det(A) into the equation: det(2A) = 4 * (-1).
Step 9: Calculate the final result: 4 * (-1) = -4.
Determinant of a Matrix – The determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties about the matrix, such as whether it is invertible.
Scaling a Matrix – When a matrix is scaled by a constant factor, the determinant of the scaled matrix is equal to the constant raised to the power of the matrix size (n) multiplied by the determinant of the original matrix.
