📚 Topic Summary
An invertible matrix, also known as a nonsingular matrix, is a square matrix that has an inverse. That is, if $A$ is an $n \times n$ matrix, then $A$ is invertible if there exists another $n \times n$ matrix $B$ such that $AB = BA = I_n$, where $I_n$ is the $n \times n$ identity matrix. Not all square matrices are invertible; a matrix is invertible if and only if its determinant is non-zero. Finding the inverse of a matrix is a fundamental operation in linear algebra with applications in solving systems of linear equations, computer graphics, and more.
This worksheet provides exercises to solidify your understanding of invertible matrices, including vocabulary, calculations, and conceptual challenges.
🧠 Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Singular Matrix |
a) A square matrix whose determinant is non-zero. |
| 2. Identity Matrix |
b) A matrix that, when multiplied by another, leaves the other matrix unchanged. |
| 3. Invertible Matrix |
c) A square matrix that does not have an inverse. |
| 4. Determinant |
d) A scalar value that can be computed from the elements of a square matrix. |
| 5. Inverse Matrix |
e) A matrix which, when multiplied by the original matrix, results in the identity matrix. |
Match each term (1-5) with its definition (a-e).
✏️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
A matrix $A$ is considered ________ if there exists a matrix $B$ such that $AB = BA = I$, where $I$ is the ________. The matrix $B$ is called the ________ of $A$. A matrix that does not have an inverse is called ________. The ________ of an invertible matrix must be non-zero.
🤔 Part C: Critical Thinking
Explain in your own words why a non-square matrix cannot be invertible.