In linear algebra, a minor of a matrix is the determinant of a smaller square matrix formed by deleting one or more rows or columns of the original matrix. A cofactor is a minor multiplied by either 1 or -1, depending on the position of the element. Cofactors are essential for calculating the determinant of a matrix, especially larger matrices, and for finding the inverse of a matrix. Let's practice these concepts!
Match the following terms with their correct definitions:
| Term | Definition |
|---|---|
| 1. Minor | A. The determinant of the submatrix after deleting row i and column j, multiplied by $(-1)^{i+j}$ |
| 2. Cofactor | B. The element in the i-th row and j-th column of a matrix. |
| 3. Matrix | C. A rectangular array of numbers, symbols, or expressions arranged in rows and columns. |
| 4. Element | D. The determinant of the submatrix formed by deleting the i-th row and j-th column. |
| 5. Determinant | E. A scalar value that can be computed from the elements of a square matrix. |
Fill in the blanks with the appropriate terms:
The ______ of an element in a matrix is found by calculating the determinant of the submatrix formed after deleting the row and column containing that element. The ______ is then found by multiplying the minor by $(-1)^{i+j}$, where i and j are the row and column indices, respectively. The determinant of a matrix can be calculated using ______ expansion.
Explain in your own words why cofactors are important when calculating the inverse of a matrix. Can you provide an example of how they are used in this process?
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