Mathematics Class 12 - Matrices And-Determinants Notes

Matrices and Determinants

Welcome to the chapter on Matrices and Determinants for Class 12. In this chapter, you will learn about matrices, their types, operations, determinants, and their applications in solving systems of equations. By the end of this chapter, you will be able to perform matrix operations, find determinants, and use them to solve mathematical problems.

Key Concepts

• Matrix: A rectangular array of numbers arranged in rows and columns.
• Order of a Matrix: The number of rows and columns in a matrix (m × n).
• Determinant: A scalar value that can be computed from a square matrix.

Types of Matrices

• Row Matrix: A matrix with only one row.
• Column Matrix: A matrix with only one column.
• Square Matrix: Number of rows = number of columns.
• Diagonal Matrix: All non-diagonal elements are zero.
• Scalar Matrix: A diagonal matrix with equal diagonal elements.
• Identity Matrix: A diagonal matrix with all diagonal elements as 1.
• Zero/Null Matrix: All elements are zero.

Matrix Operations

• Addition: Matrices of the same order can be added by adding corresponding elements.
• Subtraction: Subtract corresponding elements of matrices of the same order.
• Scalar Multiplication: Multiply every element by a scalar.
• Matrix Multiplication: Multiply rows of the first matrix by columns of the second matrix.
• Transpose: Interchange rows and columns of a matrix.

Determinants

Determinants are defined only for square matrices. The determinant of a 2×2 matrix is:

If A = | a   b |
| c   d |, then det(A) = ad - bc

For a 3×3 matrix:
If A = | a b c |
          | d e f |
          | g h i |
then det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

Properties of Determinants

• If two rows (or columns) are interchanged, the sign of the determinant changes.
• If two rows (or columns) are identical, the determinant is zero.
• If all elements of a row (or column) are multiplied by a scalar, the determinant is multiplied by that scalar.
• The determinant of a matrix is equal to the determinant of its transpose.

Adjoint and Inverse of a Matrix

• Minor: The determinant of the submatrix formed by deleting one row and one column.
• Cofactor: Minor with a sign based on position.
• Adjoint: The transpose of the cofactor matrix.
• Inverse: For a non-singular matrix A, A^-1 = adj(A)/det(A).

Applications

• Solving systems of linear equations using matrices (Matrix method, Cramer's rule).
• Representing and transforming data in computer graphics.
• Cryptography and coding theory.

Practice Questions

1. Find the sum of the matrices:
   A = | 1 2 |
   | 3 4 | and B = | 5 6 |
   | 7 8 |
2. Calculate the determinant of the matrix:
   | 2 3 |
   | 4 5 |
3. Find the transpose of the matrix:
   | 1 4 |
   | 2 5 |
   | 3 6 |
4. If det(A) = 0, is A invertible?
5. Solve the system of equations using matrices:
   x + y = 5
   2x + 3y = 12

Challenge Yourself

• Find the inverse of the matrix | 4 7 |
  | 2 6 | if it exists.
• Show that the determinant of a matrix is equal to the determinant of its transpose with an example.

Did You Know?

• Matrices are used in computer graphics to rotate and scale images.
• Determinants help in checking if a system of equations has a unique solution.

Glossary

• Matrix: A rectangular array of numbers.
• Determinant: A value calculated from a square matrix.
• Transpose: A matrix obtained by interchanging rows and columns.
• Inverse: A matrix which, when multiplied with the original, gives the identity matrix.

Answers to Practice Questions

1. A + B = | 6 8 |
   | 10 12 |
2. Determinant = (2×5) - (3×4) = 10 - 12 = -2
3. Transpose = | 1 2 3 |
   | 4 5 6 |
4. No, if det(A) = 0, A is not invertible (it is singular).
5. x = 3, y = 2

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Practice matrix operations and determinants to master advanced algebra and solve real-world problems!