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 as per the Class 12 Maths syllabus?
An invertible matrix is a square matrix 'A' for which another square matrix 'B' of the same order exists, such that their product is the identity matrix (I). This relationship is expressed as AB = BA = I. The matrix 'B' is called the inverse of 'A', denoted as A⁻¹. A matrix is invertible only if its determinant is non-zero. Invertible matrices are also known as nonsingular or nondegenerate matrices.
2. What is the main condition to check if a matrix is invertible?
The primary and most straightforward method to check if a square matrix is invertible is to calculate its determinant. If the determinant of the matrix is a non-zero value, the matrix is invertible. Conversely, if the determinant is zero, the matrix is singular (non-invertible) and does not have an inverse.
3. How is a singular matrix different from a nonsingular (invertible) matrix?
The key difference lies in their determinant and the existence of an inverse.
- An invertible (nonsingular) matrix is a square matrix with a non-zero determinant, and it has a unique inverse.
- A singular matrix is a square matrix whose determinant is exactly zero, and it does not have an inverse.
4. What are the most important properties of invertible matrices for the 2025-26 CBSE exams?
For the CBSE Class 12 exams, you should know these key properties:
- Uniqueness of Inverse: The inverse of a matrix, if it exists, is unique.
- Inverse of an Inverse: If A is invertible, then its inverse A⁻¹ is also invertible, and (A⁻¹)⁻¹ = A.
- Inverse of a Product: If A and B are invertible matrices of the same order, then their product AB is also invertible, and (AB)⁻¹ = B⁻¹A⁻¹ (This is known as the reversal law).
- Inverse of a Transpose: The inverse of the transpose of a matrix is the same as the transpose of its inverse, i.e., (Aᵀ)⁻¹ = (A⁻¹)ᵀ.
5. What is the formula to find the inverse of a matrix using the adjoint method?
The formula to find the inverse of a square matrix 'A' using the adjoint method is given by: A⁻¹ = (1/det(A)) * adj(A). Here, det(A) is the determinant of matrix A (which must be non-zero), and adj(A) is the adjugate (or adjoint) of matrix A, which is the transpose of the matrix of cofactors of the elements of A.
6. How does finding the inverse of a matrix help solve a system of linear equations?
Invertible matrices provide a powerful method to solve systems of linear equations. A system can be represented in the matrix form Ax = b, where 'A' is the matrix of coefficients, 'x' is the column matrix of variables, and 'b' is the column matrix of constants. If the coefficient matrix 'A' is invertible, you can find the unique solution for the variables by pre-multiplying both sides by A⁻¹, which gives x = A⁻¹b.
7. Why can only square matrices be invertible?
The concept of an inverse is defined by the condition AB = BA = I, where I is the identity matrix. The identity matrix itself is always a square matrix. For the matrix products AB and BA to both be defined and result in the same identity matrix, the matrices A and B must both be square and of the same order. If A were a non-square matrix (e.g., m x n where m ≠ n), its inverse B would have to be n x m. While AB would be an m x m matrix, BA would be an n x n matrix, making it conceptually impossible for AB and BA to be equal.
8. Why is the rank of an invertible n x n matrix always equal to n?
The rank of a matrix represents the number of linearly independent rows or columns. For an n x n matrix to be invertible, its determinant must be non-zero. A non-zero determinant implies that all of its rows (and columns) are linearly independent. When all 'n' rows of an n x n matrix are linearly independent, its rank is 'n'. This is why an invertible matrix is also called a full-rank matrix.
9. What is the significance of the Invertible Matrix Theorem?
The Invertible Matrix Theorem is significant because it connects many different and important concepts in linear algebra. It states that for any n x n square matrix, a list of seemingly different conditions are all logically equivalent. This means if one condition is true, all of them are true, and if one is false, all are false. These conditions include:
- The matrix is invertible.
- The determinant is non-zero.
- The rank of the matrix is n.
- The columns of the matrix are linearly independent.
- The equation Ax = 0 has only the trivial solution.
- The equation Ax = b has a unique solution for every b.
10. If the product of two matrices, AB, is the identity matrix I, does this automatically mean B is the inverse of A?
For square matrices, the answer is yes. A fundamental theorem in linear algebra states that if A and B are square matrices of the same order and AB = I, then it automatically follows that BA = I. Therefore, B is the inverse of A (and A is the inverse of B). You do not need to check the condition in both directions if the matrices are square, which simplifies the process of verifying an inverse.
