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]

Verified step by step guidance

1. Step 1: Understand the concept of inverse matrices. Two matrices A and B are inverses of each other if their product is the identity matrix. For 2x2 matrices, the identity matrix is I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.
2. Step 2: Multiply the given matrices. Let's denote the first matrix as A and the second matrix as B. Calculate the product AB by performing matrix multiplication.
3. Step 3: Perform the matrix multiplication. For matrices A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} and B = \begin{bmatrix} e & f \\ g & h \end{bmatrix}, the product AB is \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}.
4. Step 4: Check if the resulting matrix from the multiplication is the identity matrix I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.
5. Step 5: If the product AB equals the identity matrix, then the matrices are inverses of each other. If not, they are not inverses.

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.