Identity Matrix – Definition, Properties And Solved Examples - Vedantu

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

1. 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 \(I_3\). \( A \times I_3 = \begin{bmatrix} 2 & -1 & 0 \\ 0 & 3 & 4 \\ 0 & 0 & 5 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
2. Perform multiplication by row and column. Each element remains the same as in A.
3. 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