In Exercises 13 - 18, use the fact that ifa b d - bA = then A^(-1) = 1/(ad-bc) to find the inverse of c d - c aeach matrix, if possible. Check that AA^(-1) = I_2 and A^(-1)A = I_2. 3 - 1A = - 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: det(A) = ad - bc = (3)(2) - (-1)(-4).
3. If det(A) is not zero, find the inverse using the formula: A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.
4. Substitute the values into the inverse formula: A^{-1} = \frac{1}{det(A)} \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}.
5. Verify the result by checking that AA^{-1} = I_2 and A^{-1}A = I_2, where I_2 is the identity matrix.

Key Concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix A^(-1) such that when it is multiplied by the original matrix A, the result is the identity matrix I. For a 2x2 matrix, the inverse can be calculated using the formula A^(-1) = 1/(ad-bc) * [d, -b; -c, a], where 'a', 'b', 'c', and 'd' are the elements of the matrix. The determinant (ad-bc) must be non-zero for the inverse to exist.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties about the matrix. For a 2x2 matrix, the determinant is calculated as ad - bc. A non-zero determinant indicates that the matrix is invertible, while a zero determinant signifies that the matrix does not have an inverse, as it represents a singular matrix.

Identity Matrix

The identity matrix is a special type of square matrix that serves as the multiplicative identity in matrix multiplication. For a 2x2 matrix, the identity matrix I_2 is represented as [1, 0; 0, 1]. When any matrix A is multiplied by the identity matrix, the result is the original matrix A. Verifying that AA^(-1) = I_2 and A^(-1)A = I_2 confirms that A^(-1) is indeed the correct inverse of A.