Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix I￬n.) [3x3 matrix] and [3x3 matrix]

Verified step by step guidance

1. <Step 1: Understand the concept of matrix inverses. Two matrices A and B are inverses of each other if their product, in both orders (AB and BA), results in the identity matrix I.>
2. <Step 2: Multiply the first matrix by the second matrix. This involves taking the dot product of the rows of the first matrix with the columns of the second matrix.>
3. <Step 3: Multiply the second matrix by the first matrix. Again, take the dot product of the rows of the second matrix with the columns of the first matrix.>
4. <Step 4: Check if both products from Steps 2 and 3 result in the identity matrix I, which is a matrix with 1s on the diagonal and 0s elsewhere.>
5. <Step 5: Conclude whether the matrices are inverses based on whether both products are the identity 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 dimensions will be determined by the outer dimensions of the two matrices. This operation is crucial for determining if two matrices are inverses, as their product must yield the identity matrix.

Identity Matrix

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. For a 3x3 identity matrix, it looks like this: [[1, 0, 0], [0, 1, 0], [0, 0, 1]]. The identity matrix acts as a multiplicative identity in matrix algebra, meaning that any matrix multiplied by the identity matrix remains unchanged. To verify if two matrices are inverses, their product must equal the identity matrix.

Inverse Matrices

Inverse matrices are pairs of matrices that, when multiplied together, yield the identity matrix. For a matrix A, its inverse is denoted as A^-1, and the relationship is expressed as A * A^-1 = I, where I is 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 two matrices are inverses involves calculating their product and confirming it equals the identity matrix.