What is Matrices Class 12: Definition, Properties & Examples

Understanding the Definition of Matrices in Class 12

A matrix is a rectangular array of numbers arranged in rows and columns. In Class 12 NCERT Mathematics, matrices are introduced as a way to organize data and solve systems of linear equations. The size or order of a matrix is given by the number of rows and columns it contains.

For example, a matrix $A$ with 3 rows and 2 columns is written as:

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

Here, the order of matrix $A$ is $3 \times 2$ (read as "3 by 2"). Each element is denoted by $a_{ij}$ where $i$ is the row number and $j$ is the column number.

Matrices help in representing and manipulating data efficiently, which is why they are important in various fields including computer science, physics, and economics.

Types of Matrices Explained for Class 12 Students

In Class 12, you will learn about different types of matrices based on their size and elements:

• Row Matrix: Has only one row, e.g., $[3, 5, 7]$ (order $1 \times n$).
• Column Matrix: Has only one column, e.g., $\begin{bmatrix}2 \\ 4 \\ 6\end{bmatrix}$ (order $m \times 1$).
• Square Matrix: Number of rows equals number of columns, e.g., $2 \times 2$ or $3 \times 3$.
• Zero Matrix: All elements are zero.
• Identity Matrix: A square matrix with 1s on the main diagonal and 0s elsewhere.

Understanding these types helps in performing matrix operations correctly.

Matrix Operations: Addition, Subtraction, and Multiplication

Matrices can be added, subtracted, or multiplied under certain rules:

• Addition and Subtraction: Two matrices can be added or subtracted only if they have the same order. Add or subtract corresponding elements.

For example, if $$A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}, B = \begin{bmatrix}5 & 6 \\ 7 & 8\end{bmatrix}$$ then $$A + B = \begin{bmatrix}1+5 & 2+6 \\ 3+7 & 4+8\end{bmatrix} = \begin{bmatrix}6 & 8 \\ 10 & 12\end{bmatrix}$$

• Multiplication: Matrix multiplication is possible only when the number of columns in the first matrix equals the number of rows in the second matrix.

If $A$ is $m \times n$ and $B$ is $n \times p$, then the product $AB$ is an $m \times p$ matrix.

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

Then, $$AB = \begin{bmatrix}(1\times5 + 2\times7) & (1\times6 + 2\times8) \\ (3\times5 + 4\times7) & (3\times6 + 4\times8)\end{bmatrix} = \begin{bmatrix}19 & 22 \\ 43 & 50\end{bmatrix}$$

Mastering these operations is crucial for solving problems in Class 12 exams.

Determinants and Their Role in Matrices for Class 12

The determinant is a special number that can be calculated only for square matrices. It helps in understanding matrix properties such as invertibility.

• For a $2 \times 2$ matrix

$$A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ the determinant is $$|A| = ad - bc$$

• For a $3 \times 3$ matrix, the determinant is calculated using minors and cofactors.

If the determinant of a matrix is zero, the matrix is called singular and does not have an inverse. If the determinant is non-zero, the matrix is non-singular and invertible.

Example:

Calculate the determinant of $$B = \begin{bmatrix}2 & 3 \\ 1 & 4\end{bmatrix}$$

Solution: $$|B| = (2)(4) - (3)(1) = 8 - 3 = 5$$

Since $|B| \neq 0$, matrix $B$ is invertible.

Determinants are essential for solving linear equations using matrices.

Applications of Matrices in Class 12 Mathematics

Matrices are not just theoretical; they have practical applications in Class 12 and beyond:

• Solving Systems of Linear Equations: Using matrix methods like Cramer's Rule and inverse matrices.
• Computer Graphics: Representing transformations such as rotation and scaling.
• Economics: Modeling input-output tables.
• Physics and Engineering: Representing data and solving equations.

Example: Solve the system

$$\begin{cases}2x + 3y = 8 \\ 5x - y = 7\end{cases}$$

Using matrix form $AX = B$:

$$A = \begin{bmatrix}2 & 3 \\ 5 & -1\end{bmatrix}, X = \begin{bmatrix}x \\ y\end{bmatrix}, B = \begin{bmatrix}8 \\ 7\end{bmatrix}$$

Find $X = A^{-1}B$ if $A^{-1}$ exists.

This shows how matrices simplify solving equations efficiently.