Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Matrices and determinants
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Matrices and determinants
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Subject: General Mathematics
Class: Senior Secondary 3
Term: 1st Term
Week: 1
Theme: Number And Numeration
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Define matrix State the or der and notation of a matrix Mention and definetypes of matrices Perform the operation of addition,subtraction of matrices multiply matrix by as calar quantify and multiply two matrices Aand B. find the transpose of amatrise by interchanging the rowsand columns calculate the determinant of 2 x 2matrix applications to solving simultaneousequations
(3+2) ]` `[ 5 5 ]` 2.4.3 Subtraction of Matrices Matrices can be subtracted only if they have the same order. The difference is obtained by subtracting the corresponding elements. If `A = [ a_ij ]` and `B = [ b_ij ]`, then `A - B = [ a_ij - b_ij ]`.
Example: Using matrices A and B from the addition example: `A - B = [ (2-3) (1-0) ]` = `[ -1 1 ]` `[ (4-1) (3-2) ]` `[ 3 1 ]` 2.4.4 Scalar Multiplication To multiply a matrix by a scalar (a single number), multiply every element of the matrix by that scalar. If `A = [ a_ij ]` and `k` is a scalar, then `kA = [ ka_ij ]`.
Example: Given `A = [ 2 1 ]` `[ 4 3 ]` `3A = [ (32) (31) ]` = `[ 6 3 ]` `[ (34) (33) ]` `[ 12 9 ]` 2.4.5 Matrix Multiplication (Product of Two Matrices A and B) Two matrices A and B can be multiplied only if the number of columns in the first matrix (A) is equal to the number of rows in the second matrix (B). If A is an `m x p` matrix and B is a `p x n` matrix, then their product `AB` will be an `m x n` matrix. The element `c_ij` in the product matrix `C = AB` is found by multiplying the elements of the `i`-th row of A by the corresponding elements of the `j`-th column of B and summing the products.
Example: Given `A = [ 1 2 ]` and `B = [ 5 6 ]` `[ 3 4 ]` `[ 7 8 ]` (A is `2x2`, B is `2x2`. Number of columns in A (2) = Number of rows in B (2). So, multiplication is possible, and the result will be `2x2`). `AB = [ (15 + 27) (16 + 28) ]` `[ (35 + 47) (36 + 48) ]` `AB = [ (5 + 14) (6 + 16) ]` `[ (15 + 28) (18 + 32) ]` `AB = [ 19 22 ]` `[ 43 50 ]` 2.4.6 Transpose of a Matrix The transpose of a matrix A, denoted as `A^T` or `A'`, is obtained by interchanging its rows and columns. The first row of A becomes the first column of `A^T`, the second row becomes the second column, and so on. If A is an `m x n` matrix, then `A^T` will be an `n x m` matrix.
Example: Given `A = [ 1 2 3 ]` `[ 4 5 6 ]` (order `2 x 3`) `A^T = [ 1 4 ]` `[ 2 5 ]` `[ 3 6 ]` (order `3 x 2`) 2.5 Determinant of a 2x2 Matrix The determinant is a scalar value that can be computed from the elements of a square matrix. It is denoted by `det(A)` or `|A|`. For a `2x2` matrix `A = [ a b ]`, its determinant is calculated as: `[ c d ]` `det(A) = |A| = ad - bc` (the product of the elements on the main diagonal minus the product of the elements on the anti-diagonal).
Example: Given `A = [ 5 2 ]` `[ 3 4 ]` `det(A) = (5 4) - (2 * 3)` `det(A) = 20 - 6` `det(A) = 14` 2.6 Applications to Solving Simultaneous Equations (using Cramer's Rule for 2x2) Matrices provide an elegant method for solving systems of linear equations. For a system of two linear equations with two variables: `a_1x + b_1y = c_1` `a_2x + b_2y = c_2` This can be written in matrix form `AX = B`: `[ a_1 b_1 ] [ x ] = [ c_1 ]` `[ a_2 b_2 ] [ y ] [ c_2 ]` Cramer's Rule steps:
1. Calculate the determinant of the coefficient matrix `D = |A|`. `D = | a_1 b_1 | = a_1b_2 - a_2b_1` `| a_2 b_2 |`
2. Calculate `D_x` by replacing the first column (coefficients of x) in D with the constants `c_1` and `c_2`. `D_x = | c_1 b_1 | 2.1 Definition of a Matrix A matrix is a rectangular array or arrangement of numbers, symbols, or expressions, called elements or entries, arranged in rows and columns. Matrices are enclosed in brackets, either square `[ ]` or parentheses `( )`.
Example: ``` A = [ 2 -1 5 ] [ 0 3 4 ] ``` In this matrix, 2, -1, 5, 0, 3, 4 are the elements. 2.2 Order and Notation of a Matrix The order (or dimension) of a matrix describes its size by specifying the number of rows and columns it has. An `m x n` matrix has `m` rows and `n` columns. The number of rows is always stated first.
Notation: Matrices are typically denoted by capital letters (e.g., A, B, C). The elements within a matrix are denoted by corresponding lowercase letters with subscripts indicating their position (`a_ij`), where `i` represents the row number and `j` represents the column number.
Example: For matrix A above: It has 2 rows and 3 columns. So, its order is `2 x 3`. `a_11` (element in row 1, column 1) = 2 `a_12` (element in row 1, column 2) = -1 `a_23` (element in row 2, column 3) = 4 2.3 Types of Matrices
1. Row Matrix: A matrix with only one row.
Example: `[ 1 5 -2 ]` (order `1 x 3`)
2. Column Matrix: A matrix with only one column.
Example: `[ 3 ]` `[ 0 ]` `[ 7 ]` (order `3 x 1`)
3. Square Matrix: A matrix where the number of rows is equal to the number of columns (`m = n`).
Example: `[ 4 -1 ]` `[ 2 5 ]` (order `2 x 2`)
4. Rectangular Matrix: A matrix where the number of rows is not equal to the number of columns (`m ≠ n`).
Example: `[ 1 2 3 ]` `[ 4 5 6 ]` (order `2 x 3`)
5. Zero/Null Matrix: A matrix in which all elements are zero. It is usually denoted by `O`.
Example: `[ 0 0 ]` `[ 0 0 ]` (order `2 x 2`)
6. Diagonal Matrix: A square matrix where all non-diagonal elements are zero. (Diagonal elements can be zero or non-zero).
Example: `[ 5 0 ]` `[ 0 2 ]` (order `2 x 2`)
7. Identity/Unit Matrix: A square matrix in which all diagonal elements are 1 and all other elements are
0. It is denoted by `I`.
Example: `[ 1 0 ]` `[ 0 1 ]` (order `2 x 2`) `[ 1 0 0 ]` `[ 0 1 0 ]` `[ 0 0 1 ]` (order `3 x 3`) 2.4 Matrix Operations 2.4.1 Equality of Matrices Two matrices A and B are equal if and only if:
1. They have the same order.
2. Their corresponding elements are equal.
Example: If `A = [ x 3 ]` and `B = [ 2 3 ]`, then `A = B` implies `x = 2` and `y = 5`. `[ 5 y ]` `[ 5 5 ]` 2.4.2 Addition of Matrices Matrices can be added only if they have the same order. The sum is obtained by adding the corresponding elements. If `A = [ a_ij ]` and `B = [ b_ij ]`, then `A + B = [ a_ij + b_ij ]`.
Example: Given `A = [ 2 1 ]` and `B = [ 3 0 ]` `[ 4 3 ]` `[ 1 2 ]` `A + B = [ (2+3) (1+0) ]` = `[ 5 1 ]` `[ (4+1) (3+2) ]` `[ 5 5 ]` 2.4.3 Subtraction of Matrices Matrices can be subtracted only if they have the same order. The difference is obtained by subtracting the corresponding elements. If `A = [ a_ij ]` and `B = [ b_ij ]`, then `A - B = [ a_ij - b_ij ]`.
Example: Using matrices A and B from the addition example: `A - B = [ (2-3) (1-0) ]` = `[ -1 1 ]` `[ (4-1) (3-2) ]` `[ 3 1 ]` 2.4.4 Scalar Multiplication To multiply a matrix b_1y = c_1` `a_2x + b_2y = c_2` This can be written in matrix form `AX = B`: `[ a_1 b_1 ] [ x ] = [ c_1 ]` `[ a_2 b_2 ] [ y ] [ c_2 ]` Cramer's Rule steps:
1. Calculate the determinant of the coefficient matrix `D = |A|`. `D = | a_1 b_1 | = a_1b_2 - a_2b_1` `| a_2 b_2 |`
2. Calculate `D_x` by replacing the first column (coefficients of x) in D with the constants `c_1` and `c_2`. `D_x = | c_1 b_1 | = c_1b_2 - c_2b_1` `| c_2 b_2 |`
3. Calculate `D_y` by replacing the second column (coefficients of y) in D with the constants `c_1` and `c_2`. `D_y = | a_1 c_1 | = a_1c_2 - a_2c_1` `| a_2 c_2 |`
4. Solve for x and y: `x = D_x / D` `y = D_y / D` (
Note: This method works only if `D ≠ 0`).
Example: Solve the following simultaneous equations using Cramer's Rule: `2x + 3y = 7` `4x - y = 1` Step 1: Write in matrix form and find D `[ 2 3 ] [ x ] = [ 7 ]` `[ 4 -1 ] [ y ] [ 1 ]` `D = | 2 3 | = (2 -1) - (3 4) = -2 - 12 = -14` `| 4 -1 |` Step 2: Find D_x `D_x = | 7 3 | = (7 -1) - (3 1) = -7 - 3 = -10` `| 1 -1 |` Step 3: Find D_y `D_y = | 2 7 | = (2 1) - (7 * 4) = 2 - 28 = -26` `| 4 1 |` Step 4: Solve for x and y `x = D_x / D = -10 / -14 = 5 / 7` `y = D_y / D = -26 / -14 = 13 / 7` Solution: `x = 5/7`, `y = 13/7` 3.1 Introduction (10 minutes)
Teacher Activity: Initiate a discussion on how data is organized (e.g., class registers, market price lists, election results). Introduce the concept of a matrix as a structured way to represent such data. Define a matrix, its elements, rows, and columns using a simple real-life example (e.g., prices of yam, rice, and beans from two different markets in Abuja). Introduce the concept of matrix order (`m x n`) and standard notation (`a_ij`).
Student Activity: Participate in the discussion, providing examples of organized data. Observe and ask clarifying questions on the definition and notation. 3.2 Exploring Matrix Types (15 minutes)
Teacher Activity: Present various types of matrices (row, column, square, zero, identity, diagonal). For each type, provide a clear definition and a numerical example on the board. Emphasize the unique characteristics of each type, especially the identity matrix due to its importance in matrix multiplication.
Student Activity: Copy definitions and examples into their notebooks. Identify and categorize given matrices into their respective types. 3.3 Performing Basic Matrix Operations (30 minutes)
Teacher Activity: Explain the conditions for matrix equality, addition, and subtraction (same order). Demonstrate addition and subtraction step-by-step using two `2x2` or `2x3` matrices. Explain scalar multiplication and demonstrate with an example. Provide simple practice problems for students to solve on the board or in groups.
Student Activity: Attempt addition, subtraction, and scalar multiplication problems. Present solutions on the board and peer-review. 3.4 Matrix Multiplication and Transpose (40 minutes)
Teacher Activity: Crucially explain the condition for matrix multiplication: `(m x p) (p x n)` results in `(m x n)`. Walk through a detailed step-by-step example of multiplying two `2x2` matrices, highlighting the "row-by-column" process. Provide another example for a `2x3` matrix multiplied by a `3x2` matrix. Introduce the concept of matrix transpose, demonstrating how rows become columns and vice versa. Provide practice exercises for both matrix multiplication and transpose.
Student Activity: Pay close attention to the matrix multiplication process, as it is often challenging. Practice matrix multiplication and transpose with various matrix sizes. Discuss common errors and strategies for accuracy. 3.5 Determinant of 2x2 Matrices and Simultaneous Equations (40 minutes)
Teacher Activity: Define the determinant for a 2x2 matrix and provide the formula `ad - bc`. Work through several examples of calculating determinants. Introduce the application of matrices to solve simultaneous linear equations using Cramer's Rule (for 2x2 systems). Clearly demonstrate the steps for setting up the coefficient matrix, `D`, `D_x`, `D_y`, and calculating `x` and `y`. Emphasize the condition `D ≠ 0` for a unique solution.
Student Activity: Calculate determinants for given 2x2 matrices. Follow the step-by-step example of solving simultaneous equations. Attempt a new set of simultaneous equations using the matrix method. 3.6 Conclusion and Review (5 minutes)
Teacher Activity: Summarize the key concepts covered: definition, types, operations, transpose, determinant, and application to simultaneous equations. Address any lingering questions. Assign homework.
Student Activity: Ask final questions. Note down homework assignments.
Inventory Management in a Nigerian Market: A market trader at Bodija Market in Ibadan sells bags of rice, beans, and garri. They want to track their stock and sales. Matrices can be used to represent the initial stock, daily sales, and remaining stock.
Example: A `2x3` matrix could represent the prices of 3 commodities from 2 different suppliers. Matrix operations can then be used to calculate total cost, or compare prices from different suppliers.
Integration: Students can be asked to collect hypothetical prices of a few common food items from different markets and represent them as matrices, then perform basic operations like finding the difference in prices between markets. Budgeting and Financial Planning (e.g., for a family or small business): A family needs to budget for monthly expenses across different categories like food, transport, and utilities. A matrix can represent the estimated costs versus actual costs for several months.
Example: A `2x3` matrix could represent "Estimated" and "Actual" expenses for "Food," "Transport," "School Fees" over a month. Subtraction of matrices would show variances.
Integration: Students can create simple matrices for their own monthly pocket money expenses and savings, and use scalar multiplication to project yearly totals.
Resource Allocation in Agriculture: A farmer in Benue State wants to optimize the allocation of land and fertilizer for different crops (e.g., yam, maize, cassava) to maximize yield. A matrix can represent the requirements of each crop.
Example: A matrix `A` could show units of land and fertilizer needed per hectare for yam, maize, cassava. Another matrix `B` could show the available land and fertilizer. Matrix multiplication (or linear programming concepts built on matrices) helps determine optimal production.
Integration: Discuss how government agencies like the Federal Ministry of Agriculture use similar principles to plan agricultural output and resource distribution across states.