Matrices 12th Notes

📘 Class 12 Maths – Chapter 3: Matrices

🧮 Introduction and Basic Concepts (with Examples)

🔷 What is a Matrix?

A matrix is a rectangular arrangement of numbers in rows and columns.

Example:
\( \begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix} \) — Order = 2 × 3

🔶 Order of a Matrix

If a matrix has \( m \) rows and \( n \) columns, then the order is written as \( m \times n \).

\( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} \)
Order = 3 × 2

🔷 Types of Matrices

• Column Matrix: \( \begin{bmatrix} 3 \\ 5 \\ 7 \end{bmatrix} \)
• Row Matrix: \( \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \)
• Square Matrix: \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)
• Diagonal Matrix: \( \begin{bmatrix} 4 & 0 \\ 0 & 7 \end{bmatrix} \)
• Scalar Matrix: \( \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} \)
• Identity Matrix: \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
• Zero Matrix: \( \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)

🔶 Equal Matrices

Matrices are equal if they have the same order and identical corresponding elements.

\( A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \), \( B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \) ⇒ \( A = B \)

🔷 Operations on Matrices

Addition:

\( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix},\ B = \begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix} \)
\( A + B = \begin{bmatrix} 5 & 5 \\ 5 & 5 \end{bmatrix} \)

Scalar Multiplication:

\( 3 \cdot A = 3 \cdot \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix} \)

Subtraction:

\( A = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix},\ B = \begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} \)
\( A - B = \begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix} \)

Matrix Multiplication:

\( A = \begin{bmatrix} 1 & 2 \end{bmatrix},\ B = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \)
\( AB = \begin{bmatrix} 11 \end{bmatrix} \)

🔷 Transpose of a Matrix

The transpose of a matrix \( A \), denoted \( A^T \), is formed by interchanging its rows with columns.

\( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \)
\( A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} \)

🔶 Special Matrices

Symmetric Matrix:

A matrix is symmetric if \( A^T = A \)
\( A = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix} \) ✔️

Skew-Symmetric Matrix:

A matrix is skew-symmetric if \( A^T = -A \)
\( A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix} \) ✔️