The Inverse of a Matrix In Exercises 5-12, show that B is...

The Inverse of a Matrix In Exercises $5-12,$ show that $B$ is the inverse of $A .$
$$A=\left[ \begin{array}{ll}{1} & {2} \\ {3} & {4}\end{array}\right], B=\frac{1}{2} \left[ \begin{array}{rr}{-4} & {2} \\ {3} & {-1}\end{array}\right]$$

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Key Concepts

Matrix Multiplication

Matrix multiplication is the process of multiplying two matrices to produce a third matrix. This operation is essential for verifying if one matrix is the inverse of another, since the product of a matrix and its inverse should result in the identity matrix.

Matrix Inverse

The matrix inverse is a crucial concept wherein the inverse of a square matrix is another matrix such that when the two are multiplied, the result is the identity matrix. It is used extensively in solving systems of equations and transforming coordinate systems. A matrix must be non-singular, meaning it must have a nonzero determinant, in order to have an inverse.

Identity Matrix

The identity matrix is a special square matrix with ones on the main diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix operations, meaning that when any matrix is multiplied by the identity matrix, the original matrix is returned unchanged.

Determinant

The determinant is a scalar value calculated from the elements of a square matrix. It provides important information about the matrix, such as whether the matrix is invertible. A nonzero determinant indicates that the matrix has an inverse, whereas a zero determinant means the matrix is singular and does not have an inverse.

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