Identity Matrix Definition Formula Properties and Solved Examples
The concept of identity matrix plays a key role in mathematics, especially in matrix algebra and systems of equations. Understanding identity matrices makes it easy to solve, manipulate, and invert matrices, which is essential for board exams and competitive tests.
What Is Identity Matrix?
An identity matrix is a square matrix in which all the elements along the main diagonal are 1, and all other elements are 0. It is often denoted by I or In (where n is the matrix order). You’ll find this concept applied in areas such as matrix multiplication, matrix inversion, and solving equations using matrices in maths and computer science.
Key Formula for Identity Matrix
Here’s the standard formula for an n × n identity matrix:
\( I_n = \begin{bmatrix} 1 & 0 & 0 & \dots & 0 \\ 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \end{bmatrix} \)
| Order | Identity Matrix Example |
|---|---|
| 2×2 | \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \) |
| 3×3 | \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \) |
| 4×4 | \( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) |
Cross-Disciplinary Usage
The identity matrix is not only useful in Maths but also plays an important role in Physics, Computer Science, Engineering, and logical reasoning. Students preparing for JEE or NEET will see its relevance while learning about transformations, system-solving, and algorithms.
Step-by-Step Illustration
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Given a 3×3 matrix \(A = \begin{bmatrix} 2 & -1 & 0 \\ 0 & 3 & 4 \\ 0 & 0 & 5 \end{bmatrix}\), multiply by the 3×3 identity matrix \(I_3\).
\( A \times I_3 = \begin{bmatrix} 2 & -1 & 0 \\ 0 & 3 & 4 \\ 0 & 0 & 5 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
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Perform multiplication by row and column.
Each element remains the same as in A.
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The product is \(A\) itself:
\( A \times I_3 = A \)
Speed Trick or Vedic Shortcut
Here’s a quick check: To instantly recognize an identity matrix in an MCQ, scan the main diagonal entries—if they are all 1 and the matrix is square (rows = columns), it’s the identity. This saves time in exams when spotting answers fast.
Example Trick: For a 4×4 matrix, if any off-diagonal entry is nonzero or any diagonal entry ≠ 1, it’s NOT an identity matrix.
Try These Yourself
- Write the 5×5 identity matrix.
- Check if \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{bmatrix} \) is an identity matrix.
- If \( B \) is a 2×2 matrix, what is \( B \times I_{2} \)?
- Find the order of an identity matrix with 6 diagonal entries.
Frequent Errors and Misunderstandings
- Assuming any diagonal matrix is an identity matrix (only the main diagonal should be ones, others zero).
- Forgetting that identity matrices must be square matrices.
- Mixing up the identity matrix with the zero matrix.
Relation to Other Concepts
The idea of identity matrix connects closely with topics such as types of matrices (like diagonal and zero matrices) and matrix inversion. Mastering this helps with understanding equation solving, determinants, and advanced algebra.
Classroom Tip
A quick way to remember the identity matrix: Imagine it as the "number 1" of matrices—multiplying any matrix by it leaves the matrix unchanged. Vedantu’s teachers often use the phrase "identity keeps things the same" to help you recall this during live classes and mock tests.
We explored the identity matrix—from its definition, formula, visual structure, calculation steps, and mistakes to useful connections. Keep practicing problems using identity matrices and join live sessions on Vedantu for even deeper mastery of this and related matrix operations.
Inverse Matrix | Types of Matrices | Zero Matrix | Properties of Matrices Inverse
FAQs on Identity Matrix in Linear Algebra Explained Clearly
1. What is an identity matrix?
An identity matrix is a square matrix with 1s on the main diagonal and 0s everywhere else. It is usually denoted by I or In for an n × n matrix.
- For a 2 × 2 matrix: I2 = [[1, 0], [0, 1]]
- For a 3 × 3 matrix: I3 has 1s at positions (1,1), (2,2), (3,3)
- It acts as the multiplicative identity in matrix multiplication.
2. What is the order of an identity matrix?
The order of an identity matrix is always n × n, meaning it is a square matrix. The number of rows equals the number of columns.
- I2 is 2 × 2
- I3 is 3 × 3
- In general, In has n rows and n columns
3. Why is the identity matrix important in matrix multiplication?
The identity matrix is important because multiplying any matrix by it leaves the matrix unchanged. For any n × n matrix A, A × I = I × A = A.
- It behaves like the number 1 in regular multiplication.
- It preserves the original matrix.
- This property defines it as the multiplicative identity.
4. How do you multiply a matrix by an identity matrix?
To multiply a matrix by an identity matrix, perform normal matrix multiplication and the result will be the original matrix. If A is 2 × 2 and I2 is the identity matrix, then A × I2 = A.
- Example: A = [[2, 3], [4, 5]]
- I2 = [[1, 0], [0, 1]]
- Result: [[2, 3], [4, 5]]
5. What is the difference between an identity matrix and a zero matrix?
The key difference is that an identity matrix has 1s on the main diagonal, while a zero matrix has all entries equal to 0.
- Identity matrix: Used as multiplicative identity.
- Zero matrix: Used as additive identity.
- Example: I2 = [[1,0],[0,1]] vs O = [[0,0],[0,0]]
6. What is the determinant of an identity matrix?
The determinant of any identity matrix is always 1. For any size n, det(In) = 1.
- Example: det(I2) = (1×1 − 0×0) = 1
- This holds true for all square identity matrices.
7. What are the properties of an identity matrix?
The identity matrix has several key properties in linear algebra.
- A × I = I × A = A (Multiplicative identity)
- det(I) = 1
- I-1 = I (Inverse of identity is itself)
- It is always a square matrix.
8. What is the inverse of an identity matrix?
The inverse of an identity matrix is the identity matrix itself. Mathematically, I-1 = I.
- Because I × I = I
- The identity matrix already satisfies the inverse condition A × A-1 = I
9. Can you give an example of a 3×3 identity matrix?
A 3 × 3 identity matrix has 1s on the main diagonal and 0s elsewhere. It is written as:
- I3 = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
10. How is the identity matrix used in solving matrix equations?
The identity matrix is used to solve matrix equations by finding inverses, especially in equations like AX = B. If A is invertible, then:
- Multiply both sides by A-1
- A-1AX = A-1B
- Since A-1A = I, we get X = A-1B
