How to Check if a Matrix is Invertible?
The concept of invertible matrix plays a key role in mathematics and is widely applicable to solving equations, computer graphics, cryptography, and various competitive exams. Mastering this topic helps students quickly decide if a given matrix can be “reversed” or “undone” in calculations.
What Is an Invertible Matrix?
An invertible matrix (also called a nonsingular matrix) is a square matrix that has an inverse. The inverse is another matrix which, when multiplied with the original, returns the identity matrix. You’ll find this concept applied in areas such as matrix algebra, systems of linear equations, and determinants.
Key Formula for Invertible Matrix
Here’s the golden rule: a matrix \( A \) is invertible if there exists a matrix \( B \) such that \( AB = BA = I \), where \( I \) is the identity matrix of the same order. The main test for invertibility is: the determinant must not be zero.
2 × 2 Example:
For \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), its inverse exists if \( ad-bc \neq 0 \).
Formula for the inverse: \( A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \)
Determinant and Invertibility
The determinant plays the central role when checking for an invertible matrix. If determinant \( \neq 0 \), the matrix is invertible. If determinant = 0, it’s called a singular or non-invertible matrix. The same rule applies for 2×2, 3×3, or any n×n square matrices.
Step-by-Step Illustration
- Check if the matrix is square (same number of rows and columns).
Only square matrices can be invertible. - Calculate the determinant.
If 2×2: \( ad-bc \); If 3×3: use expansion (can refer to determinant of 3x3 matrix). - If the determinant ≠ 0, the matrix is invertible.
- If the determinant = 0, it’s non-invertible (singular).
Invertible Matrix Theorem — Quick Table
| Equivalent Conditions (A is n×n) | Meaning |
|---|---|
| A is invertible | An inverse matrix exists |
| det A ≠ 0 | Determinant is non-zero |
| A is row/column equivalent to I | Can be reduced to identity |
| Ax = 0 has only trivial solution x=0 | Columns of A are linearly independent |
| rank A = n | Full rank; no zero eigenvalues |
Properties and Examples
Important properties:
- If A is invertible, so is \( A^{-1} \) and \( (A^{-1})^{-1} = A \).
- The product of two invertible matrices is invertible, and \( (AB)^{-1} = B^{-1}A^{-1} \).
- Inverse of transpose: \( (A^T)^{-1} = (A^{-1})^T \)
- If scalar \( c \neq 0 \), then \( (cA)^{-1} = \frac{1}{c}A^{-1} \)
Example 1: 2×2
Let \( M = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)
1. Compute det M: \( 1\times4 - 3\times2 = 4 - 6 = -2 \neq 0 \)
2. Therefore, M is invertible.
Example 2: 3×3
Let \( N = \begin{bmatrix} 2 & 1 & 3 \\ 0 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \)
1. det N ≠ 0 (can use Laplace expansion).
2. Therefore, N is invertible.
You can see detailed determinant calculations and solved steps at this Vedantu study page.
Non-Invertible (Singular) Matrix
A singular matrix is a square matrix whose determinant is zero. Such matrices are called non-invertible because no matrix exists that can “reverse” their effect.
| Type | Determinant | Invertibility |
|---|---|---|
| Invertible (Nonsingular) | ≠ 0 | Has an inverse |
| Non-invertible (Singular) | 0 | No inverse exists |
See more about this in Singular Matrix and Diagonal Matrix articles.
Speed Trick or Vedic Shortcut
Need to check invertibility fast? For any 2×2 matrix, simply compute \( ad-bc \); if not zero, move on. For larger matrices, students use row reduction: if you can reach the identity matrix, it’s invertible! Save time by double-checking zeros in any entire row or column (means determinant is zero, non-invertible).
Speed tips like these are part of Vedantu’s live Maths classes for exam-focused learning.
Try These Yourself
- Is \( \begin{bmatrix} 2 & 6 \\ 1 & 3 \end{bmatrix} \) invertible?
- If det \( A = 0 \), what type of matrix is A?
- For \( \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix} \), check its invertibility and find the inverse if possible.
- Give one example of a 3×3 non-invertible matrix.
Frequent Errors and Misunderstandings
- Forgetting that only square matrices can be invertible.
- Thinking a zero determinant means inverse exists (it doesn’t).
- Mixing up identity and singular matrices.
Relation to Other Concepts
The idea of an invertible matrix connects with the identity matrix, matrix multiplication, square matrices, and the matrix inverse. Mastering invertibility leads to better handling of linear equations and transformations in higher studies.
Classroom Tip
A quick way to remember invertibility: “If you can go back to the start (identity matrix), your path (matrix) is invertible!” Vedantu’s teachers use stories and simple visuals to help you lock in the invertible matrix concept.
We explored invertible matrices—from the definition, formula, key steps, mistakes, and how it links across maths topics. Continue practicing with Vedantu’s study resources to become confident in matrices, determinants, and algebra. Invertible matrix problems appear across the curriculum, so keep sharpening your skills!
Read more at Inverse Matrix and broaden your Maths strengths by exploring Types of Matrices and Determinants and Matrices on Vedantu.
FAQs on Invertible Matrix: Definition, Properties, and Solved Examples
1. What is an invertible matrix?
An invertible matrix, also known as a nonsingular or nondegenerate matrix, is a square matrix that has an inverse. This means there exists another matrix, its inverse, such that when multiplied with the original matrix, the result is the identity matrix. A square matrix is invertible if and only if its determinant is non-zero.
2. How do I check if a matrix is invertible?
The primary method to determine if a square matrix is invertible is to calculate its determinant. If the determinant is non-zero, the matrix is invertible; if the determinant is zero, the matrix is singular (non-invertible). You can also use row reduction to check if the matrix is row equivalent to the identity matrix.
3. What is the relationship between the determinant and invertibility?
A square matrix is invertible if and only if its determinant is non-zero. The determinant acts as a crucial test for invertibility. A zero determinant indicates a singular matrix, which lacks an inverse.
4. What are some properties of invertible matrices?
Several key properties define invertible matrices: * The inverse of an invertible matrix is unique. * The product of two invertible matrices is also invertible. * The transpose of an invertible matrix is also invertible. * If a matrix A is invertible, then its inverse (A-1) is also invertible, and (A-1)-1 = A. * If c is a non-zero scalar and A is invertible, then cA is also invertible.
5. What is a singular matrix, and how does it differ from an invertible matrix?
A singular matrix, also called a non-invertible matrix, is a square matrix whose determinant is zero. Unlike invertible matrices, singular matrices do not possess an inverse. This means there is no matrix that, when multiplied by the singular matrix, yields the identity matrix.
6. How are invertible matrices used in solving systems of linear equations?
Invertible matrices are fundamental to solving systems of linear equations. If a system can be represented in matrix form as Ax = b, where A is an invertible coefficient matrix, then the unique solution is given by x = A-1b. The invertibility of A guarantees a unique solution.
7. What is the invertible matrix theorem?
The invertible matrix theorem provides a collection of equivalent conditions that determine whether a square matrix is invertible. These conditions involve concepts such as row equivalence to the identity matrix, linear independence of columns, non-zero determinant, and the existence of a unique solution to the equation Ax = b for all vectors b.
8. How do I find the inverse of an invertible matrix?
Methods for finding the inverse of an invertible matrix include using the adjugate method (for smaller matrices) or using row reduction (Gaussian elimination) to transform the augmented matrix [A|I] into [I|A-1]. Calculators and software packages also provide efficient tools for matrix inversion.
9. Are all square matrices invertible?
No, not all square matrices are invertible. Only square matrices with a non-zero determinant are invertible. Matrices with a determinant of zero are singular and do not have an inverse.
10. What are some real-world applications of invertible matrices?
Invertible matrices have numerous applications, including: * Solving systems of linear equations in various fields such as engineering and physics. * Cryptography, for encoding and decoding messages. * Computer graphics and image processing, for transformations and rotations in 3D space. * Economics and finance, for modeling complex systems.
11. What happens if I try to find the inverse of a non-invertible matrix?
Attempting to find the inverse of a non-invertible (singular) matrix will result in a failure of the inversion process. Row reduction will lead to a row of zeros, and the determinant will be zero, indicating the non-existence of an inverse.
12. Can a non-square matrix be invertible?
No, only square matrices can be invertible. The concept of an inverse matrix requires the matrix to be square (same number of rows and columns) to allow for multiplication resulting in the identity matrix.
