Verify that (adj A) transpose = (adj A transpose) for A=[111 12-3 2-13]

We are tasked with verifying that:

\[
(\text{adj}(A))^T = \text{adj}(A^T)
\]

For the given matrix:

\[
A = \begin{bmatrix}
1 & 1 & 1 \\
1 & 2 & -3 \\
2 & -1 & 3
\end{bmatrix}
\]

### Step 1: Calculate the transpose of \(A\)

The transpose of \(A\), denoted \(A^T\), is obtained by swapping the rows and columns of \(A\):

\[
A^T = \begin{bmatrix}
1 & 1 & 2 \\
1 & 2 & -1 \\
1 & -3 & 3
\end{bmatrix}
\]

### Step 2: Compute the adjugate of \(A\) and \(A^T\)

The adjugate of a matrix is the transpose of its cofactor matrix. To find the adjugate, we first compute the cofactor matrix for both \(A\) and \(A^T\).

#### Cofactor matrix of \(A\)
The cofactor matrix is composed of the determinants of the 2x2 minors obtained by removing the row and column of each element.

For \(A\):

- Minor corresponding to \(a_{11}\) (removing row 1, column 1):
  \[
  \begin{vmatrix}
  2 & -3 \\
  -1 & 3
  \end{vmatrix}
  = (2)(3) - (-3)(-1) = 6 - 3 = 3
  \]

- Minor corresponding to \(a_{12}\):
  \[
  \begin{vmatrix}
  1 & -3 \\
  2 & 3
  \end{vmatrix}
  = (1)(3) - (-3)(2) = 3 + 6 = 9
  \]

- Minor corresponding to \(a_{13}\):
  \[
  \begin{vmatrix}
  1 & 2 \\
  2 & -1
  \end{vmatrix}
  = (1)(-1) - (2)(2) = -1 - 4 = -5
  \]

Similarly, the rest of the minors can be computed. Let's proceed with calculating the adjugate for both matrices and confirming the relation.

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Here are 5 related questions to deepen understanding:
1. What is the adjugate matrix and how is it related to the inverse of a matrix?
2. How are cofactors of a matrix calculated?
3. Why is the transpose of the adjugate matrix important in linear algebra?
4. Can we always compute the adjugate of a singular matrix?
5. What is the difference between the adjugate and the determinant of a matrix?

**Tip:** The adjugate of a matrix is key to finding its inverse; the inverse of a matrix \(A\) is given by \(A^{-1} = \frac{1}{\det(A)} \text{adj}(A)\), provided \(\det(A) \neq 0\).

Verify that (adj A) transpose = (adj A transpose) for A=[111 12-3 2-13]

This problem involves verifying that the transpose of the adjugate of matrix A equals the adjugate of the transpose of A. It explores key linear algebra concepts such as adjugate matrices, cofactors, and matrix transposition. The step-by-step solution is suitable for undergraduate-level students studying linear algebra.

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