In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I

Verified step by step guidance

1. insert step 1: Identify the matrix A for which you need to find the inverse.
2. insert step 2: Calculate the determinant of matrix A. If the determinant is zero, the matrix does not have an inverse.
3. insert step 3: If the determinant is non-zero, find the adjugate (or adjoint) of matrix A.
4. insert step 4: Use the formula A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) to find the inverse of matrix A.
5. insert step 5: Verify your result by checking that both AA^{-1} = I and A^{-1}A = I, where I is the identity matrix.

Key Concepts

Inverse of a Matrix

The inverse of a matrix A, denoted A^(-1), is a matrix that, when multiplied by A, yields the identity matrix I. This means that AA^(-1) = I and A^(-1)A = I. Not all matrices have inverses; a matrix must be square and have a non-zero determinant to possess an inverse.

Identity Matrix

The identity matrix, denoted I, is a special square matrix that has ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication, meaning that for any matrix A, multiplying by I leaves A unchanged (AI = A and IA = A). The identity matrix is crucial for verifying the correctness of matrix inverses.

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. A non-zero determinant indicates that the matrix is invertible, while a zero determinant signifies that the matrix does not have an inverse. Understanding determinants is essential for determining whether a matrix can be inverted.