Quantitative Analysis for Management, 13/e

M5.3 Finding the Inverse of a Matrix

The inverse of a matrix is a unique matrix of the same dimensions that, when multiplied by the original matrix, produces a unit or identity matrix. For example, if A is any $2\times 2$ matrix and its inverse is denoted $A^{-1},$ then

$$A\times A^{-1}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=\text{Identity} \text{matrix}$$ (M5-3)

The adjoint of a matrix is extremely helpful in forming the inverse of the original matrix. We simply compute the value of the determinant of the original matrix and divide each term of the adjoint by this value.

To find the inverse of the matrix just presented, we need to know the adjoint (already computed) and the value of the determinant of the original matrix:

$$\left(\begin{array}{ccc}3& 7& 5\\ 2& 0& 3\\ 4& 1& 8\end{array}\right)=\text{Original} \text{matrix}$$

Determining the value of the determinant.

Value of determinant: ...