Matrices | Class 12 Maths Chapter 3 Notes

📌 Introduction to Matrices

Matrices form one of the most important topics in Class 12 Mathematics and play a crucial role in higher studies, especially in engineering, computer science, economics, and physics. A matrix is essentially a rectangular arrangement of numbers, symbols, or expressions in rows and columns.

Matrices are used to represent and solve systems of linear equations, perform transformations, and model real-world data efficiently.

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📖 Definition of Matrix

A matrix is a rectangular array of elements arranged in rows and columns, enclosed within brackets.

A matrix is usually denoted by capital letters like A, B, C, etc.

Example:$$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$A=[14​25​36​]

Here:

• Rows = 2
• Columns = 3

So, the order of matrix = 2 × 3

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📊 Order of a Matrix

The order of a matrix is defined as:$$\text{Order} = \text{Number of Rows} \times \text{Number of Columns}$$Order=Number of Rows×Number of Columns

Example:

• If a matrix has 3 rows and 2 columns → Order = 3 × 2

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🔢 Types of Matrices

1. Row Matrix

A matrix having only one row.

Example:$$[1 \quad 2 \quad 3]$$[123]

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2. Column Matrix

A matrix having only one column.

Example:$$\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$$​123​​

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3. Rectangular Matrix

When number of rows ≠ number of columns.

Example:
2 × 3 matrix

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4. Square Matrix

When number of rows = number of columns.

Example:$$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$[13​24​]

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5. Zero Matrix (Null Matrix)

All elements are zero.

Example:$$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$[00​00​]

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6. Diagonal Matrix

All non-diagonal elements are zero.

Example:$$\begin{bmatrix} 2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix}$$​200​050​007​​

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7. Scalar Matrix

Diagonal elements are equal.

Example:$$\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$$[30​03​]

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8. Identity Matrix (Unit Matrix)

All diagonal elements are 1.$$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$I=[10​01​]

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9. Upper Triangular Matrix

Elements below diagonal are zero.

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10. Lower Triangular Matrix

Elements above diagonal are zero.

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🔄 Equality of Matrices

Two matrices A and B are equal if:

1. They have the same order
2. Corresponding elements are equal

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➕ Addition of Matrices

Two matrices can be added only if they have the same order.

If$$A = [a_{ij}], \quad B = [b_{ij}]$$A=[aij​],B=[bij​]

Then$$A + B = [a_{ij} + b_{ij}]$$A+B=[aij​+bij​]

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➖ Subtraction of Matrices

$$A – B = [a_{ij} – b_{ij}]$$A−B=[aij​−bij​]

Same rule: order must be same.

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✖️ Multiplication of Matrix by Scalar

If k is a scalar and A is a matrix:$$kA = [k \cdot a_{ij}]$$kA=[k⋅aij​]

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✖️ Multiplication of Matrices

Matrix multiplication is not as simple as numbers.

Condition:
If A is m × n and B is n × p, then AB is possible.

Result:
AB will be of order m × p.

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Formula:

$$(AB)_{ij} = \sum a_{ik} \cdot b_{kj}$$(AB)ij​=∑aik​⋅bkj​

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Important Points:

• Matrix multiplication is not commutative $$AB \neq BA$$AB=BA
• But it is:
  • Associative
  • Distributive

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🔁 Transpose of a Matrix

Transpose means converting rows into columns.

If$$A = [a_{ij}]$$A=[aij​]

Then$$A^T = [a_{ji}]$$AT=[aji​]

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Properties of Transpose

1. $(A^T)^T = A$(AT)T=A
2. $(A + B)^T = A^T + B^T$(A+B)T=AT+BT
3. $(AB)^T = B^T A^T$(AB)T=BTAT

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🔷 Symmetric and Skew-Symmetric Matrices

Symmetric Matrix

$$A^T = A$$AT=A

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Skew-Symmetric Matrix

$$A^T = -A$$AT=−A

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🧮 Elementary Operations (Transformations)

These are operations used to simplify matrices.

Types:

1. Row interchange
2. Row multiplication
3. Row addition

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🔁 Elementary Matrices

These are obtained by performing a single elementary operation on identity matrix.

Used in finding inverse of matrices.

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🔑 Inverse of a Matrix

A square matrix A has an inverse if:$$A \cdot A^{-1} = I$$A⋅A−1=I

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Conditions:

• Matrix must be square
• Determinant ≠ 0

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📐 Formula for Inverse (2×2 Matrix)

$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$A=[ac​bd​] $$A^{-1} = \frac{1}{ad – bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$A−1=ad−bc1​[d−c​−ba​]

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⚠️ Singular and Non-Singular Matrix

• Singular → Determinant = 0
• Non-Singular → Determinant ≠ 0

Only non-singular matrices have inverse.

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🧩 Solving System of Linear Equations

Matrices are widely used to solve equations like:$$ax + by = c$$ax+by=c $$dx + ey = f$$dx+ey=f

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Matrix Form:

$$AX = B$$AX=B

Where:

• A = coefficient matrix
• X = variable matrix
• B = constant matrix

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Solution:

$$X = A^{-1} B$$X=A−1B

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📊 Applications of Matrices

Matrices are used in:

• Computer graphics
• Cryptography
• Economics
• Physics
• Engineering
• Data science
• Machine learning

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📌 Important Properties of Matrices

1. Commutative Law (Addition only)

$$A + B = B + A$$A+B=B+A

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2. Associative Law

$$(A + B) + C = A + (B + C)$$(A+B)+C=A+(B+C)

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3. Distributive Law

$$A(B + C) = AB + AC$$A(B+C)=AB+AC

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4. Identity Law

$$AI = IA = A$$AI=IA=A

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❗ Common Mistakes to Avoid

• Multiplying matrices without checking order
• Assuming AB = BA
• Forgetting transpose properties
• Miscalculating determinant

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🧠 Tips for Exams

• Always write order of matrix
• Practice multiplication carefully
• Remember inverse formula
• Solve step-by-step

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📚 Practice Questions

1. Find transpose of given matrix
2. Multiply two matrices
3. Find inverse of 2×2 matrix
4. Solve linear equations using matrices

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🎯 Conclusion

Matrices are a powerful mathematical tool that simplifies complex calculations and systems. Mastering this chapter helps build a strong foundation for higher mathematics and competitive exams like JEE, NDA, and CUET.

Understanding concepts like matrix operations, transpose, inverse, and applications ensures clarity and confidence in problem-solving.