Lesson Objectives
• Learn how to find the adjoint of a matrix

## Finding the Adjoint of a Matrix

In this lesson, we will learn how to find the adjoint of a matrix. The adjoint of a matrix, which is also referred to as the classical adjoint or the adjugate is found using two easy but tedious steps.

• Create a matrix of cofactors
• Find the transpose of the matrix of cofactors
Let's look at an example.
Example #1: Find the adjoint of A. $$A=\left[ \begin{array}{ccc}-4&1&-1\\ -3&-1&-1\\3&2&3\end{array}\right]$$ First, let's create a matrix of cofactors, we will call this matrix C. $$C=\left[ \begin{array}{ccc}-1&6&-3\\ -5&-9&11\\-2&-1&7\end{array}\right]$$ Now, we will find the transpose of this matrix. We will use the notation CT. $$C^T=\left[ \begin{array}{ccc}-1&-5&-2\\ 6&-9&-1\\-3&11&7\end{array}\right]$$ Since the transpose of the cofactor matrix is the adjoint, we can replace our CT with adj(A). $$adj(A)=\left[ \begin{array}{ccc}-1&-5&-2\\ 6&-9&-1\\-3&11&7\end{array}\right]$$

#### Skills Check:

Example #1

Find the adjoint of A. $$A=\left[ \begin{array}{ccc}3&-2&2\\ -2&2&1\\2&2&-5\end{array}\right]$$

A
$$\left[ \begin{array}{ccc}1&6&8\\ 4&3&5\\ 6&14&8\end{array}\right]$$
B
$$\left[ \begin{array}{ccc}3&5&7\\ -6&-3&4\\ 7&8&6\end{array}\right]$$
C
$$\left[ \begin{array}{ccc}-1&-5&-2\\ -3&11&7\\ 6&5&8\end{array}\right]$$
D
$$\left[ \begin{array}{ccc}-12&-6&-6\\ -8&-19&-7\\ -8&-10&2\end{array}\right]$$
E
$$\left[ \begin{array}{ccc}5&2&7\\ 5&-3&4\\5&6&3\end{array}\right]$$

Example #2

Find the adjoint of A. $$A=\left[ \begin{array}{ccc}1&5&-7\\ 8&6&-2\\-1&-3&8\end{array}\right]$$

A
$$\left[ \begin{array}{ccc}-3&7&6\\ -6&9&4\\4&23&17\end{array}\right]$$
B
$$\left[ \begin{array}{ccc}2&8&11\\ 4&-1&-1\\-2&5&9\end{array}\right]$$
C
$$\left[ \begin{array}{ccc}42&-19&32\\ -62&1&-54\\-18&-2&-34\end{array}\right]$$
D
$$\left[ \begin{array}{ccc}-1&-13&-2\\ 14&7&-3\\-3&11&7\end{array}\right]$$
E
$$\left[ \begin{array}{ccc}-1&-5&5\\ 6&3&-1\\-3&22&7\end{array}\right]$$