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. 10 - 2A = - 5 1

Verified step by step guidance

1. Identify the elements of matrix A: a = 10, b = -2, c = -5, d = 1.
2. Calculate the determinant of A: det(A) = ad - bc = (10)(1) - (-2)(-5).
3. Check if the determinant is non-zero to ensure the inverse exists.
4. Use the formula for the inverse: A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.
5. Substitute the values into the inverse formula: A^{-1} = \frac{1}{det(A)} \begin{bmatrix} 1 & 2 \\ 5 & 10 \end{bmatrix}.

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 A = [a b; c d], the inverse exists if the determinant (ad - bc) is non-zero. The formula for the inverse is A^(-1) = 1/(ad-bc) * [d -b; -c a].

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 A = [a b; c d], the determinant is calculated as ad - bc. A non-zero determinant indicates that the matrix is invertible, while a zero determinant means it is singular and does not have an inverse.

Identity Matrix

The identity matrix is a special type of square matrix that acts as the multiplicative identity in matrix multiplication. For a 2x2 matrix, the identity matrix I_2 is represented as [1 0; 0 1]. When a matrix A is multiplied by its inverse A^(-1), the result is the identity matrix, confirming that A^(-1) is indeed the correct inverse.