Determinants are scalar values computed from square matrices. They're super useful for solving linear equations, finding inverses, and understanding matrix properties. Some key properties include: if you swap two rows, the determinant changes sign; if you multiply a row by a scalar, the determinant is multiplied by that scalar; and if you add a multiple of one row to another, the determinant remains unchanged. Understanding these properties can simplify determinant calculations and solve matrix-related problems faster! 🧮
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Determinant | A. A square array of numbers. |
| 2. Matrix | B. The value obtained by specific calculations on a square matrix. |
| 3. Scalar Multiplication | C. Multiplying a matrix by a constant. |
| 4. Row Operation | D. Adding a multiple of one row to another. |
| 5. Transpose | E. Switching rows and columns of a matrix. |
Answers: 1-B, 2-A, 3-C, 4-D, 5-E
Complete the following sentences using the correct terms.
If you interchange two rows of a matrix, the _______ of the determinant changes. If a matrix has a row of zeros, its determinant is _______. If a row is multiplied by a scalar $k$, the determinant is multiplied by _______. Adding a multiple of one row to another _______ the determinant.
Answers: sign, zero, $k$, doesn't change
Explain how the properties of determinants can be used to determine if a matrix is invertible. Provide an example.
Answer: A matrix is invertible if and only if its determinant is non-zero. For example, consider the matrix $A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$. The determinant of A is $(2*4) - (1*3) = 8 - 3 = 5$, which is non-zero. Therefore, matrix A is invertible. If the determinant was zero, the matrix would not be invertible. 💡
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