How to Find Inverse of an Invertible Matrix with Formula and Examples
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 Explained with Concept and Conditions
1. What is an invertible matrix?
An invertible matrix is a square matrix that has another matrix called its inverse, such that their product equals the identity matrix. If a matrix A is invertible, then there exists a matrix A-1 satisfying:
A · A-1 = A-1 · A = I
Key points:
- Only square matrices can be invertible.
- The identity matrix I acts like the number 1 in matrix multiplication.
- Invertible matrices are also called non-singular matrices.
2. How do you know if a matrix is invertible?
A matrix is invertible if and only if its determinant is not equal to zero. For a square matrix A:
det(A) ≠ 0
Equivalent conditions include:
- The matrix has full rank.
- The rows and columns are linearly independent.
- The matrix can be reduced to the identity matrix using row operations.
3. What is the formula for the inverse of a 2×2 matrix?
The inverse of a 2×2 matrix A = [[a, b], [c, d]] is given by:
A-1 = (1/(ad − bc)) [[d, −b], [−c, a]], provided that ad − bc ≠ 0.
Steps:
- Compute the determinant: det(A) = ad − bc.
- Swap a and d.
- Change the signs of b and c.
- Multiply the result by 1/det(A).
4. Can you give an example of finding the inverse of a matrix?
Yes, for A = [[2, 1], [5, 3]], the inverse exists because its determinant is non-zero.
Step 1: Compute determinant:
det(A) = (2×3 − 1×5) = 6 − 5 = 1
Step 2: Apply formula:
A-1 = (1/1) [[3, −1], [−5, 2]]
Final answer:
A-1 = [[3, −1], [−5, 2]]
5. What is the difference between a singular and an invertible matrix?
The difference is that an invertible (non-singular) matrix has a non-zero determinant, while a singular matrix has a determinant equal to zero.
Comparison:
- Invertible matrix: det(A) ≠ 0, inverse exists.
- Singular matrix: det(A) = 0, no inverse exists.
- Singular matrices have dependent rows or columns.
6. Why must an invertible matrix be square?
An invertible matrix must be square because only square matrices can produce the identity matrix when multiplied by another matrix.
If A is m×n and has an inverse, then:
- A · A-1 = I must be defined.
- This requires m = n.
7. How do you find the inverse of a 3×3 matrix?
The inverse of a 3×3 matrix is found using the adjugate method or row reduction, provided the determinant is non-zero.
Using the adjugate method:
- Compute det(A).
- Find the matrix of cofactors.
- Transpose it to get the adjugate matrix.
- Multiply by 1/det(A).
A-1 = (1/det(A)) adj(A)
8. What are the properties of an invertible matrix?
An invertible matrix satisfies several important algebraic properties.
Key properties:
- (A-1)-1 = A
- (AB)-1 = B-1A-1
- (AT)-1 = (A-1)T
- det(A-1) = 1/det(A)
9. How is an invertible matrix used to solve linear equations?
An invertible matrix is used to solve a system of linear equations written as AX = B by multiplying both sides by A-1.
Solution method:
- Given AX = B
- Multiply both sides by A-1
- X = A-1B
10. What happens if the determinant of a matrix is zero?
If the determinant of a matrix is zero, the matrix is singular and does not have an inverse.
Consequences:
- The rows or columns are linearly dependent.
- The matrix cannot be reduced to the identity matrix.
- A system AX = B may have no solution or infinitely many solutions.
