What Is Matrix Class 12 Maths: Definition, Properties & Examples

Definition of Matrix in Class 12 Maths

In Class 12 NCERT Maths, a matrix is defined as a rectangular array of numbers arranged in rows and columns. Each number in the matrix is called an element. If a matrix has $m$ rows and $n$ columns, it is called an $m \times n$ matrix. For example:

$$ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{bmatrix} $$

Here, matrix $A$ has 2 rows and 3 columns, so its order is $2 \times 3$. Matrices are usually denoted by capital letters such as $A$, $B$, or $M$.

Types of Matrices and Their Properties

Class 12 students should be familiar with different types of matrices, each with unique properties:

• Row Matrix: Has only one row, e.g., $[1, 2, 3]$.
• Column Matrix: Has only one column, e.g., $\begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix}$.
• Square Matrix: Number of rows equals number of columns, e.g., $3 \times 3$ matrix.
• Zero Matrix: All elements are zero.
• Diagonal Matrix: Square matrix with nonzero elements only on the main diagonal.
• Scalar Matrix: Diagonal matrix with all diagonal elements equal.
• Identity Matrix: Scalar matrix with diagonal elements equal to 1.
• Symmetric Matrix: Square matrix where $A = A^T$ (transpose).

Basic Operations on Matrices

Understanding matrix operations is crucial for Class 12 exams. The main operations include:

• Addition: Two matrices $A$ and $B$ of the same order can be added by adding corresponding elements.

$$ (A + B)_{ij} = A_{ij} + B_{ij} $$

• Subtraction: Similar to addition, subtract corresponding elements.

• Scalar Multiplication: Multiply each element by a scalar $k$.

$$ (kA)_{ij} = k \times A_{ij} $$

• Matrix Multiplication: If $A$ is $m \times n$ and $B$ is $n \times p$, then the product $AB$ is $m \times p$ where

$$ (AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj} $$

Worked Example:

Multiply

$$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} $$

Solution:

$$ AB = \begin{bmatrix} (1)(5)+(2)(7) & (1)(6)+(2)(8) \\ (3)(5)+(4)(7) & (3)(6)+(4)(8) \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} $$

Determinant and Inverse of a Matrix

The determinant is a scalar value that can be computed only for square matrices. It helps determine if a matrix is invertible.

For a $2 \times 2$ matrix:

$$ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad \det(A) = ad - bc $$

If $\det(A) \neq 0$, then $A$ is invertible and its inverse is:

$$ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} $$

Worked Example:

Find the determinant and inverse of

$$ A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix} $$

Solution:

$$ \det(A) = (4)(6) - (7)(2) = 24 - 14 = 10 $$

Since $\det(A) \neq 0$, inverse exists:

$$ A^{-1} = \frac{1}{10} \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{bmatrix} $$

Applications of Matrices in Class 12 Maths

Matrices are not just theoretical; they have practical applications in Class 12 Mathematics, including:

• Solving Systems of Linear Equations: Using matrix methods like Cramer's rule and inverse matrix method.
• Transformations: Representing geometric transformations such as rotations and reflections.
• Computer Graphics: Matrices help in image processing and 3D modelling.
• Economics and Statistics: Organising data in tabular form for analysis.

For example, to solve the system:

$$ \begin{cases} 2x + 3y = 5 \\ 4x - y = 11 \end{cases} $$

Represent as matrix equation:

$$ AX = B, \quad A = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}, X = \begin{bmatrix} x \\ y \end{bmatrix}, B = \begin{bmatrix} 5 \\ 11 \end{bmatrix} $$

If $A^{-1}$ exists, then

$$ X = A^{-1}B $$

This method is faster and more reliable for larger systems.