Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix I￬n.) [2x2 matrix] and [2x2 matrix]

Key Concepts

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce a third matrix. For two matrices A and B to be multiplied, the number of columns in A must equal the number of rows in B. The resulting matrix's elements are calculated by taking the dot product of the rows of A with the columns of B. Understanding this operation is crucial for determining if two matrices are inverses.

Identity Matrix

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. For a 2x2 matrix, the identity matrix is represented as I = [[1, 0], [0, 1]]. When a matrix A is multiplied by its inverse A⁻¹, the result is the identity matrix. This property is essential for verifying if two matrices are inverses of each other.

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. Not all matrices have inverses; a matrix must be square and have a non-zero determinant to possess an inverse. Checking if the product of two matrices equals the identity matrix is the primary method for confirming their inverse relationship.