Bihar Board - Class 12 Math - Chapter 3: Matrices Handwritten Notes

Matrices are rectangular arrays of numbers arranged in rows and columns. They are used to represent and solve systems of linear equations, transformations in geometry, and other mathematical structures. This chapter introduces the concept of matrices, their types, operations, and properties. Matrices play a crucial role in areas such as computer graphics, physics, economics, and engineering.

Key Points:

1. Definition of a Matrix:

   • A matrix is a rectangular arrangement of numbers or elements in rows and columns. The size of a matrix is defined by its number of rows and columns, expressed as $$m×nm \times n (m rows and n columns).
   • Example: $$$$A=(123456)A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} This matrix has 2 rows and 3 columns (2x3 matrix).

2. Types of Matrices:

   • Row Matrix: A matrix with only one row (e.g., $$1×n1 \times n matrix).
   • Column Matrix: A matrix with only one column (e.g., $$m×1m \times 1 matrix).
   • Square Matrix: A matrix with the same number of rows and columns (e.g., $$n×nn \times n).
   • Zero or Null Matrix: A matrix where all the elements are zero.
   • Diagonal Matrix: A square matrix where all elements outside the diagonal are zero.
   • Scalar Matrix: A diagonal matrix where all the diagonal elements are the same.
   • Identity Matrix: A square matrix with ones on the diagonal and zeros elsewhere.
   • Transposed Matrix: A matrix obtained by swapping rows and columns of a given matrix.

3. Matrix Operations:

   • Addition of Matrices: Two matrices can be added if they have the same dimensions, by adding their corresponding elements.
   • Scalar Multiplication: Multiplying each element of the matrix by a scalar (real number).
   • Matrix Multiplication: Matrices can be multiplied when the number of columns in the first matrix is equal to the number of rows in the second matrix. The result is a matrix with the number of rows of the first matrix and the number of columns of the second matrix.
   • Transpose of a Matrix: The transpose of a matrix is obtained by interchanging its rows and columns.
   • Determinant and Inverse: The determinant of a square matrix provides important information about the matrix. A matrix has an inverse if and only if its determinant is non-zero. The inverse is used to solve systems of linear equations.

4. Properties of Matrices:

   • Matrices satisfy several properties like associativity, distributivity, and commutative property for addition, but matrix multiplication is generally not commutative (i.e., $$AB≠BAAB \neq BA).
   • Identity matrix and inverse matrix have special properties that simplify calculations in solving linear systems.

5. Application of Matrices:

   • Solving Linear Equations: Matrices are widely used in solving systems of linear equations using methods like Gaussian elimination or Cramer's rule.
   • Transformations: Matrices are used to represent geometric transformations such as rotation, scaling, and translation in computer graphics.
   • Economics and Engineering: Matrices model systems in economics (input-output models) and engineering (structural analysis).

Conclusion:

Matrices are a powerful tool in mathematics with wide-ranging applications in various fields such as physics, economics, engineering, and computer science. Understanding their types, operations, and properties is crucial for solving complex systems of linear equations and performing transformations. Mastery of matrix operations like addition, multiplication, and finding the inverse opens the door to solving real-world problems efficiently.