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.