Advanced Topics: Adjoint and Cramer's Rule

1. The Adjoint Matrix and Finding Inverses

We learned the Gauss-Jordan algorithm to find the inverse of a large matrix. There is also a determinant-based formula to find the inverse using the Adjoint.

Definition

Let $A$ be a square matrix, and let $C$ be its corresponding cofactor matrix (the matrix formed by replacing every entry in $A$ with its cofactor).

The adjoint of $A$, denoted by $\text{adj}(A)$, is the transpose of the cofactor matrix:

The Adjoint Inverse Formula

If $A$ is an invertible matrix, then you can find its inverse by multiplying the adjoint by $1$ over the determinant:

How to Use It

1. Calculate the determinant of $A$. (If it's $0$, stop—it's not invertible!)
2. Find the cofactor for every single entry to build matrix $C$.
3. Transpose $C$ to get $\text{adj}(A)$.
4. Multiply $\text{adj}(A)$ by the scalar $\frac{1}{\det(A)}$.

2. Cramer's Rule

Cramer's Rule is a fantastic shortcut for solving a linear system without ever doing row reduction! It uses determinants to solve directly for individual variables.

The Theorem

Let $AX = b$ be a system of $n$ equations in $n$ unknowns such that $\det(A) \neq 0$. Then the system has a unique solution given by the formulas:

How to find $A_j$

The matrix $A_j$ is obtained by taking the original coefficient matrix $A$, and completely replacing its $j$-th column with the constant vector $b$.

Example

Solve the system:

$$\begin{cases} x - 3y + z = 4 \\ 2x - y = -2 \\ 4x - 3z = 0 \end{cases}$$

Step 1 - Find $\det(A)$:

Evaluate the determinant of the coefficient matrix. Let's say $\det(A) = -11$.

Step 2 - Find $x$:

Replace the 1st column of $A$ with the constants $(4, -2, 0)$. Evaluate the determinant of this new matrix $A_1$:

Step 3 - Find $y$:

Replace the 2nd column of $A$ with the constants $(4, -2, 0)$. Evaluate the determinant to get $\det(A_2)$:

Step 4 - Find $z$:

Replace the 3rd column of $A$ with the constants $(4, -2, 0)$. Evaluate the determinant to get $\det(A_3)$: