Grade 11 Maths Unit 3: 3.4 Elementary Raw Operations of Matrices Exercise 3.11, 3.12 & 3.13| Saquama

Introduction

This tutorial provides a step-by-step guide to performing elementary raw operations with matrices, based on Exercises 3.11, 3.12, and 3.13 from Grade 11 Maths Unit 3. Understanding these operations is essential for solving various mathematical problems, particularly in algebra and linear equations.

Step 1: Understanding Matrices

• A matrix is a rectangular array of numbers arranged in rows and columns.
• Each element in a matrix is identified by its position, with the first index representing the row and the second index representing the column.

Key Matrix Types

• Row Matrix: A matrix with a single row.
• Column Matrix: A matrix with a single column.
• Square Matrix: A matrix with the same number of rows and columns.

Step 2: Matrix Addition

• To add two matrices, they must have the same dimensions (same number of rows and columns).
• Procedure:
  1. Align the matrices so that corresponding elements are in the same position.
  2. Add the corresponding elements together.

Example

If

A = | 1  2 |
    | 3  4 |

B = | 5  6 |
    | 7  8 |

Then

A + B = | 1+5  2+6 | = | 6  8 |
         | 3+7  4+8 |   | 10 12 |

Step 3: Matrix Subtraction

• Similar to addition, matrix subtraction requires the matrices to have the same dimensions.
• Procedure:
  1. Align the matrices.
  2. Subtract the corresponding elements.

Example

If

A = | 5  6 |
    | 7  8 |

B = | 1  2 |
    | 3  4 |

Then

A - B = | 5-1  6-2 | = | 4  4 |
         | 7-3  8-4 |   | 4  4 |

Step 4: Scalar Multiplication

• Scalar multiplication involves multiplying each element of a matrix by a constant (scalar).
• Procedure:
  1. Choose a scalar value.
  2. Multiply every element of the matrix by this scalar.

Example

If

k = 2
A = | 1  2 |
    | 3  4 |

Then

k * A = | 2*1  2*2 | = | 2  4 |
         | 2*3  2*4 |   | 6  8 |

Step 5: Matrix Multiplication

• To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.
• Procedure:
  1. Take the dot product of rows from the first matrix with columns of the second matrix.
  2. Place the results in the corresponding position in the resulting matrix.

Example

If

A = | 1  2 |
    | 3  4 |

B = | 5  6 |
    | 7  8 |

Then

A * B = | (1*5 + 2*7)  (1*6 + 2*8) | = | 19  22 |
         | (3*5 + 4*7)  (3*6 + 4*8) |   | 43  50 |

Step 6: Common Pitfalls

• Ensure matrices match in size when performing addition or subtraction.
• Check dimensions for multiplication validity.
• Remember that matrix multiplication is not commutative; i.e., A * B is not necessarily equal to B * A.

Conclusion

Understanding elementary raw operations with matrices is fundamental in mathematics. This guide covered addition, subtraction, scalar multiplication, and multiplication of matrices. Practice these operations with different matrices to reinforce your understanding. For further learning, you may explore additional exercises or related topics in linear algebra.