Determine whether each pair of matrices are inverses of each...

Determine whether each pair of matrices are inverses of each other.
$$
A=\left[\begin{array}{cc}{2} & {-1} \\ {1} & {-3}\end{array}\right], B=\left[\begin{array}{rr}{\frac{1}{2}} & {0} \\ {0} & {-\frac{1}{3}}\end{array}\right]
$$

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Key Concepts

Matrix Multiplication

Matrix multiplication is an operation that takes two matrices and produces another matrix. The product of matrices involves a dot product of the rows of the first matrix with the columns of the second matrix. Correctly computing this product is essential to verify if two matrices are inverses, since the product must equal the identity matrix to confirm the inverse relationship.

Inverse Matrix

An inverse matrix is one that, when multiplied by the original matrix, results in the identity matrix. This is analogous to the concept of a reciprocal for numbers, where the inverse undoes the effect of the original matrix. If A is an n×n matrix, then its inverse, denoted as A?¹, satisfies the conditions AA?¹ = A?¹A = I?, where I? is the identity matrix of order n.

Identity Matrix

The identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. It acts as the multiplicative identity in matrix multiplication, meaning that for any matrix A, multiplying by the identity matrix I yields A, i.e., AI = IA = A.

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