📚 Topic Summary
Finding the inverse of a 2x2 matrix is a crucial skill in linear algebra. Given a matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, its inverse, denoted as $A^{-1}$, exists if and only if its determinant ($ad - bc$) is non-zero. The inverse is then calculated as $A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$. This worksheet provides practice to solidify this concept.
🧠 Part A: Vocabulary
Match each term with its definition:
- Matrix
- Determinant
- Inverse Matrix
- Scalar
- Identity Matrix
Definitions:
- A rectangular array of numbers, symbols, or expressions arranged in rows and columns.
- A special square matrix that, when multiplied by another matrix, leaves the other matrix unchanged.
- A single number used for multiplication.
- A number associated with a square matrix that reveals properties of the matrix and can be used to solve systems of linear equations.
- A matrix which, when multiplied by the original matrix, results in the identity matrix.
| Term |
Definition |
| Matrix |
a |
| Determinant |
d |
| Inverse Matrix |
e |
| Scalar |
c |
| Identity Matrix |
b |
✍️ Part B: Fill in the Blanks
The inverse of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is found by first calculating the __________. If this value is non-zero, the inverse exists. The inverse is then calculated as $A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$, where we swap $a$ and $d$, negate $b$ and $c$, and multiply by one over the __________. If the determinant is __________, the matrix is singular and does not have an inverse.
Possible answers: determinant, determinant, zero
🤔 Part C: Critical Thinking
Explain in your own words why a matrix with a determinant of zero does not have an inverse. What does this imply about the system of equations the matrix might represent?