Question

Use the classical adjoint to find \(\mathrm{A}^{-1}\) where $$ \begin{array}{rlr} \mathrm{A}= & \mid 1 & 0 & -1 \\ & \mid 0 & 2 & 2 \\ & \mid 1 & 1 & -1 \end{array} $$

Short answer

The inverse of matrix A using the classical adjoint method is: $$ A^{-1} = \begin{pmatrix} 2 & -0.5 & 0 \\ 1 & 0 & -1.5 \\ 1 & -0.5 & -1 \end{pmatrix} $$

Step-by-step solution

Find the cofactor matrix of A

To find the cofactor matrix, replace each entry in A with its cofactor. A cofactor is found by taking the determinant of the matrix that remains after removing the row and column containing the entry. Here is how we calculate each entry for the cofactor matrix: Cofactor of \(a_{11} (C_{11}):\) Remove the 1st row and 1st column: $$ C_{11} = \begin{vmatrix} 2 & 2 \\ 1 & -1 \end{vmatrix} = (2)(-1) - (2)(1) = -4 $$ Cofactor of \(a_{12} (C_{12}):\) Remove the 1st row and 2nd column: $$ C_{12} = \begin{vmatrix} 0 & 2 \\ 1 & -1 \end{vmatrix} = (0)(-1) - (2)(1) = -2 $$ Cofactor of \(a_{13} (C_{13}):\) Remove the 1st row and 3rd column: $$ C_{13} = \begin{vmatrix} 0 & 2 \\ 1 & 1 \end{vmatrix} = (0)(1) - (2)(1) = -2 $$ Cofactor of \(a_{21} (C_{21}):\) Remove the 2nd row and 1st column: $$ C_{21} = \begin{vmatrix} 0 & -1 \\ 1 & -1 \end{vmatrix} = (0)(-1) - (-1)(1) = 1 $$ Cofactor of \(a_{22} (C_{22}):\) Remove the 2nd row and 2nd column: $$ C_{22} = \begin{vmatrix} 1 & -1 \\ 1 & -1 \end{vmatrix} = (1)(-1) - (1)(-1) = 0 $$ Cofactor of \(a_{23} (C_{23}):\) Remove the 2nd row and 3rd column: $$ C_{23} = \begin{vmatrix} 1 & 0 \\ 1 & 1 \end{vmatrix} = (1)(1) - (0)(1) = 1 $$ Cofactor of \(a_{31} (C_{31}):\) Remove the 3rd row and 1st column: $$ C_{31} = \begin{vmatrix} 0 & -1 \\ 0 & 2 \end{vmatrix} = (0)(2) - (-1)(0) = 0 $$ Cofactor of \(a_{32} (C_{32}):\) Remove the 3rd row and 2nd column: $$ C_{32} = \begin{vmatrix} 1 & -1 \\ 1 & 2 \end{vmatrix} = (1)(2) - (1)(-1) = 3 $$ Cofactor of \(a_{33} (C_{33}):\) Remove the 3rd row and 3rd column: $$ C_{33} = \begin{vmatrix} 1 & 0 \\ 0 & 2 \end{vmatrix} = (1)(2) - (0)(0) = 2 $$ Now, we can construct the cofactor matrix: $$ C = \begin{pmatrix} -4 & -2 & -2 \\ 1 & 0 & 1 \\ 0 & 3 & 2 \end{pmatrix} $$

Transpose the cofactor matrix to get the adjoint

To find the adjoint of A, simply transpose the cofactor matrix: $$ adj(A) = C^T = \begin{pmatrix} -4 & 1 & 0 \\ -2 & 0 & 3 \\ -2 & 1 & 2 \end{pmatrix} $$

Compute the determinant of A

To find the inverse, we need to multiply the adjoint by the reciprocal of the determinant of A. Let's compute the determinant of A: \[ \begin{aligned} det(A) &= \begin{vmatrix} 1 & 0 & -1 \\ 0 & 2 & 2 \\ 1 & 1 & -1 \end{vmatrix} \\ &= 1 \begin{vmatrix} 2 & 2 \\ 1 & -1 \end{vmatrix} - 0 \begin{vmatrix} 0 & 2 \\ 1 & -1 \end{vmatrix} - (-1) \begin{vmatrix} 0 & 2 \\ 1 & 1 \end{vmatrix} \\ &= 1((-2)-(2)) + 2 \\ &= -4 + 2 \\ &= -2 \end{aligned} \]

Multiply the adjoint by the reciprocal of the determinant

Finally, we can find the inverse of A by multiplying the adjoint by the reciprocal of the determinant: $$ A^{-1} = \frac{1}{det(A)} \cdot adj(A) = \frac{1}{-2} \begin{pmatrix} -4 & 1 & 0 \\ -2 & 0 & 3 \\ -2 & 1 & 2 \end{pmatrix} = \begin{pmatrix} 2 & -0.5 & 0 \\ 1 & 0 & -1.5 \\ 1 & -0.5 & -1 \end{pmatrix} $$ The inverse of matrix A using the classical adjoint method is: $$ A^{-1} = \begin{pmatrix} 2 & -0.5 & 0 \\ 1 & 0 & -1.5 \\ 1 & -0.5 & -1 \end{pmatrix} $$

Key concepts

Cofactor Matrix

A cofactor matrix is crucial when calculating the inverse of a matrix using the classical adjoint method. The process begins with identifying the cofactor for each element within the matrix.

Here's how it works: For each element of a matrix, you remove the row and column that the element is in, leaving behind a smaller 2x2 matrix. The determinant of this smaller matrix is known as the minor. The cofactor is then computed using the formula \( C_{ij} = (-1)^{i+j} \cdot M_{ij} \), where \( M_{ij} \) is the minor determinant for the element at position \((i, j)\).

The signs of the cofactors alternate in a checkerboard pattern starting with a positive sign in the top-left. Thus, the cofactor matrix is formed by taking these values and placing them in the same position as their corresponding element in the original matrix. Creating a correct cofactor matrix is an essential step, as any error will propagate through to the calculation of the inverse matrix.

Adjoint Matrix

Once you have the cofactor matrix, obtaining the adjoint matrix involves a simple process: transposing the cofactor matrix. Transposition means flipping a matrix over its main diagonal, which transforms each row into a column, and each column into a row.

Why transpose? This rearranges the cofactors in a way that sets up the matrix correctly for further calculations toward finding the inverse.

The adjoint matrix, also known as the adjugate, is vital for solving the inverse as it directly influences the resulting calculations. It's a transition step that prepares you for determining the inverse but doesn't change the numerical values themselves, just their positions within the matrix.

Determinant

To compute the inverse of a matrix, calculating the determinant is a non-negotiable step. The determinant of matrix \( A \) serves as a scalar value and provides insights into many properties of the matrix, such as whether an inverse exists. A determinant equal to zero indicates a singular matrix which doesn't have an inverse.

For our 3x3 matrix \( A \), the determinant is found using the formula:
\[ det(A) = a_{11}(a_{22}a_{33} - a_{32}a_{23}) - a_{12}(a_{21}a_{33} - a_{31}a_{23}) + a_{13}(a_{21}a_{32} - a_{31}a_{22}) \]

This formula involves several minors and cofactor calculations, giving us a clearer picture of how to apply determinants within larger square matrices. In our example, \( det(A) = -2 \), allowing us to proceed with calculating the inverse matrix because it's not zero, confirming the inverse exists.

Inverse Matrix

The inverse matrix, denoted as \( A^{-1} \), is a fundamental concept in linear algebra. It's the matrix that, when multiplied with the original matrix \( A \), returns the identity matrix. We find the inverse using the classical adjoint method:

- Calculate the cofactor matrix.
- Transpose to obtain the adjoint.
- Find the determinant of the original matrix.
- Multiply the adjoint by the reciprocal of the determinant.

Mathematically, this is expressed as:
\[ A^{-1} = \frac{1}{det(A)} \cdot adj(A) \]

For our matrix \( A \), with a determinant of \(-2\), the inverse comes from multiplying and scaling the adjoint by \(-\frac{1}{2}\). Through this approach, the resulting matrix \( A^{-1} \) showcases how interconnected these steps are, further demonstrating their necessity and order in matrix algebra. Understanding these procedures makes matrix inversion more intuitive and accessible.