To determine if a matrix is invertible, follow these steps:

### 1. **Check if the matrix is square**
   - A matrix must be **square** (same number of rows and columns) in order to be invertible. If the matrix is not square, it is **not invertible**.
   - Example: A \( 3 \times 3 \) or \( 2 \times 2 \) matrix can potentially be invertible, but a \( 3 \times 2 \) matrix cannot.

### 2. **Calculate the Determinant**
   - For a square matrix \( A \), calculate the **determinant** \( \det(A) \).
   - If the determinant is **non-zero**, then the matrix is **invertible**.
   - If the determinant is **zero**, then the matrix is **not invertible** (it's **singular**).
   
   Example:
   \[
   \text{For a 2x2 matrix} \quad A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad \det(A) = ad - bc
   \]
   
   If \( \det(A) = 0 \), the matrix is not invertible.

### 3. **Check the Rank of the Matrix**
   - The matrix is invertible if its **rank** is equal to the size of the matrix.
     - For an \( n \times n \) matrix, if the rank is \( n \), it is invertible.
     - If the rank is less than \( n \), it is not invertible.
   
   To check the rank:
   - Perform Gaussian elimination or row reduction to bring the matrix to its row echelon form (REF).
   - The number of non-zero rows (pivot positions) will give the rank.

### 4. **Use the Inverse Formula for Small Matrices**
   - For a \( 2 \times 2 \) matrix, you can explicitly calculate the inverse if the determinant is non-zero using the formula:
     \[
     A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
     \]
   - For \( 3 \times 3 \) matrices, calculating the inverse requires using **adjugates** and determinants, but you can also check its invertibility by calculating the determinant first.

### 5. **Check if the Matrix Has Full Rank**
   - A matrix is invertible if the row vectors or column vectors are linearly independent. If the rows or columns are dependent (i.e., some rows or columns can be written as a linear combination of others), the matrix is singular and not invertible.

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Would you like to go into more detail about calculating determinants or row-reducing matrices? Here are 5 related questions:

1. How do you calculate the determinant for a \( 3 \times 3 \) matrix?
2. Can a matrix with repeated rows or columns be invertible?
3. What is the relationship between the rank of a matrix and its invertibility?
4. How do you perform Gaussian elimination to check if a matrix is invertible?
5. Can you explain how the adjugate method is used to find the inverse of a matrix?

**Tip:** When checking the invertibility of large matrices, instead of manually calculating the determinant, use computational tools or software (like MATLAB or Python with NumPy) to speed up the process.

Learn how to check if a matrix is invertible, including essential steps like checking if it's square, calculating its determinant, and evaluating its rank. This guide explains the concept of invertibility in linear algebra and provides examples for 2x2 and larger matrices. Ideal for high school students studying linear algebra.

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