📚 Topic Summary
The Adjoint method provides a way to calculate the inverse of a matrix using its adjoint (or adjugate) and determinant. For a square matrix $A$, its inverse, denoted as $A^{-1}$, can be found using the formula:
$A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)$
Where $\text{det}(A)$ is the determinant of $A$, and $\text{adj}(A)$ is the adjoint of $A$. The adjoint is the transpose of the cofactor matrix of $A$. This method is particularly useful for smaller matrices and provides a direct approach to finding the inverse.
🧮 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Adjoint of a Matrix |
A. The value obtained by summing the products of elements of a row (or column) of a matrix with their corresponding cofactors. |
| 2. Determinant |
B. The matrix obtained by transposing the cofactor matrix of a given matrix. |
| 3. Cofactor |
C. A scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. |
| 4. Transpose |
D. The determinant of the submatrix formed by removing the $i$-th row and $j$-th column from the original matrix, multiplied by $(-1)^{i+j}$. |
| 5. Minor |
E. A matrix formed by interchanging the rows and columns of a given matrix. |
✍️ Part B: Fill in the Blanks
The inverse of a matrix $A$ can be found using the adjoint method if the _________ is non-zero. The formula to find the inverse is $A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)$, where $\text{adj}(A)$ is the _________ of the cofactor matrix. Each element of the cofactor matrix is found using _________ and the sign convention. If the determinant equals zero, the matrix is said to be _________ and does not have an inverse.
🤔 Part C: Critical Thinking
Explain in your own words why the determinant of a matrix must be non-zero for the inverse to exist. What does a zero determinant tell you about the matrix and its corresponding system of linear equations?