In Exercises 13 - 18, use the fact that if a b d - b A = then A^(-1) = 1/(ad-bc) to find the inverse of c d - c a each matrix, if possible. Check that AA^(-1) = I_2 and A^(-1)A = I_2. 3 - 1 A = - 4 2

Verified step by step guidance

1. Identify the elements of matrix A: a = 3, b = -1, c = -4, d = 2.
2. Calculate the determinant of A using the formula det(A) = ad - bc, which is (3)(2) - (-1)(-4).
3. Check if the determinant is not zero; if it is zero, the inverse does not exist.
4. Use the formula for the inverse matrix: A^(-1) = (1/det(A)) * [[d, -b], [-c, a]], substituting the values of a, b, c, and d.
5. Multiply matrix A by its inverse A^(-1) and verify that the product equals the 2x2 identity matrix I_2, confirming the correctness of the inverse.

Key Concepts

Matrix Inverse

The inverse of a square matrix A, denoted A⁻¹, is a matrix that when multiplied by A yields the identity matrix I. For a 2x2 matrix, the inverse exists only if the determinant (ad - bc) is non-zero. The inverse is calculated using the formula involving swapping and negating elements, divided by the determinant.

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix A = [[a, b], [c, d]] is computed as ad - bc. It is a scalar value that indicates whether the matrix is invertible; if the determinant is zero, the matrix has no inverse. The determinant also provides information about the matrix's scaling factor and orientation.

Verification of Matrix Inverse

To confirm that a matrix B is the inverse of A, multiply A by B and B by A. Both products should equal the identity matrix I. This verification ensures the correctness of the inverse calculation and confirms that the inverse matrix satisfies the definition.