Determine whether or not the given pairs of matrices are...

Determine whether or not the given pairs of matrices are inverse pairs. [HINT: See Quick Examples $1-3 .]$
$$
A=\left[\begin{array}{ll}
2 & 0 \\
0 & 3
\end{array}\right], B=\left[\begin{array}{ll}
\frac{1}{2} & 0 \\
0 & \frac{1}{2}
\end{array}\right]
$$

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Key Concepts

Matrix Multiplication

Matrix multiplication is a binary operation that combines two matrices to produce a third matrix. This process involves taking the dot product of rows from the first matrix with columns from the second matrix. In the context of finding an inverse, matrix multiplication is used to check whether the product of two matrices is the identity matrix, confirming they are inverse pairs.

Matrix Inverse

A matrix inverse is a matrix that, when multiplied with the original matrix, produces the identity matrix. For a square matrix A, if there exists a matrix B such that A multiplied by B equals the identity matrix (and vice versa), then B is the inverse of A. This relationship is fundamental in solving systems of linear equations and in various other operations in linear algebra.

Identity Matrix

The identity matrix is a special square matrix that has ones on its main diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication, meaning that any square matrix multiplied by the identity matrix remains unchanged. This property is crucial for verifying the correctness of an inverse matrix.

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