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Unit 2: Algebra

Introduction to Matrices

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. A matrix having \(m\) rows and \(n\) columns is called a matrix of order \(m \times n\).

Types of Matrices

  1. Column Matrix: A matrix having only one column.
  2. Row Matrix: A matrix having only one row.
  3. Square Matrix: A matrix in which the number of rows is equal to the number of columns (\(m = n\)).
  4. Diagonal Matrix: A square matrix where all non-diagonal elements are zero.

Algebra of Matrices

Addition and Subtraction

Two matrices can be added or subtracted if and only if they have the same order.

\[ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \]

\[ A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix} \]

Multiplication

The product of two matrices \(A\) and \(B\) is defined if the number of columns of \(A\) is equal to the number of rows of \(B\).

Matrix A m × n × Matrix B n × p = Matrix C m × p Inner dimensions must match Outer dimensions form the new matrix

Determinants and Inverses

For any square matrix \(A\) of order \(n\), we associate a number (real or complex) called its determinant, denoted by \(|A|\) or \(\det(A)\).

A square matrix \(A\) is invertible if and only if \(A\) is a non-singular matrix (\(|A| \neq 0\)).

\[ A^{-1} = \frac{1}{|A|} \text{adj}(A) \]

System of Linear Equations

Using matrices, we can solve systems of simultaneous linear equations with the form \(AX = B\), giving the solution \(X = A^{-1}B\).

Competency-Based Questions

Q1. A firm produces three products, \(P_1\), \(P_2\), and \(P_3\), which require raw materials \(R_1\) and \(R_2\). Formulate a matrix representation showing the raw material requirements for producing 100 units of each product.

Q2. Suppose the supply chain function of a company can be transformed using the operation \( AB = C \). If \( A \) is a \( 3 \times 2 \) matrix representing supply routes and \( B \) is a \( 2 \times 4 \) matrix representing capacities, determine the dimensions of the output matrix \(C\) and interpret its meaning in a real-world scenario.

Q3. Solve the following system of linear equations using the matrix method to deduce the equilibrium prices for three commodities: \[ 2x + y + z = 1 \ x - 2y - z = \frac{3}{2} \ 3y - 5z = 9 \]

Q4. A school needs to allocate resources equally among its sports, arts, and science departments based on a matrix of their budgets. Explain why an inverse matrix might not exist for the budget allocation matrix if all departments exactly mirror the budget ratios of the principal fund.

Answers

Ans 1. Let \( R \) be the \( 2 \times 3 \) matrix of raw materials per product. Then requirement matrix \( M = R \times \begin{bmatrix} 100 \ 100 \ 100 \end{bmatrix} \).

Ans 2. \( C \) is a \( 3 \times 4 \) matrix. In a supply chain, it might represent the total supply capacity distribution taking into account 3 sets of routes to 4 different processing or warehouse nodes.

Ans 3. The determinant \( |A| = -17 \). Using \( X = A^{-1}B \), the solution resolves to \( x=1, y=2, z=-3 \). (Assuming typical equilibrium problem, standard inverses apply).

Ans 4. If the departments mirror the principal fund’s budget ratios, the rows of the budget matrix will be proportional (linearly dependent). This makes the determinant zero, meaning the matrix is singular and its inverse does not exist.