Delhi College of Arts and Commerce Subject: Business Mathematics and Statistics Class: B.Com (P), Sem- 2, Sec- A

Business Mathematics Chapter- Matrices and , a set of numbers arranged in rows and columns so as to form a rectangular array. The numbers are called the elements, or entries, of the matrix. If there are m rows and n columns, the matrix is said to be an “m by n” matrix, written “m × n.” For example,

is a 2 × 3 matrix. A matrix with n rows and n columns is called a of order n.

An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1 represents the element at the second row and first column of the matrix. Types of Matrices 1. Row matrix- A matrix with one row. Its size is 1 × n. Example:

2. Column Matrix- A matrix with one column. It’s size is n × 1.

Example:

3. Square Matrix- A matrix with the same number of rows and columns. It’s size is n × n

Example:

Basic Operations of a Matrix

1. Algebra of Matrices Let A and B are two matrices of same order. Then the addition of A and B denoted by A+B is the matrix obtained by adding corresponding entries of A and B. Example:

2. Multiplication of Matrices Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. Example:

Example:

3. Multiplication- Let A be an m x n matrix and K be a real or a complex number (called a scalar). Then, the multiplication of A by K, denoted by KA is obtained by multiplying each entry of A by K. This operation is called .

4. of a Matrix- The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa. In short, the transpose Transpose is a matrix formed by swapping the rows into columns and vice-versa.

What are Determinants

The of a matrix is a special number that can be calculated from a square matrix.

For a 2×2 Matrix

For a 2×2 matrix (2 rows and 2 columns):

The determinant is:

|A| = ad − bc "The determinant of A equals a times d minus b times c"

Example:

A Matrix (This one has 2 Rows and 2 Columns)

The determinant of that matrix is (calculations are explained later):

3×6 − 8×4 = 18 − 32 = −14

Minor of a determinant A minor is defined as a value computed from the determinant of a square matrix which is obtained after crossing out a row and a column corresponding to the element that is under consideration. th th Minor of an element aij of a determinant is the determinant obtained by deleting its i row and j column in which element aij lies. Minor of an element aij is denoted by Mij.

Cofactor of a determinant

The cofactor is defined as the signed minor. Cofactor of an element aij, denoted by Aij is defined by i+j A = (–1) M, where M is minor of aij. Note

• We note that if the sum i+j is even, then Aij = Mij, and that if the sum is odd, then Aij = −Mij. • Hence, the only difference between the related minor entries and cofactors may be a sign change or nothing at all.

• Whether or Aij = Mij or Aij = −Mij

Example

Find the minors and cofactors of all the elements of the determinant

Solution: Minor of the element aij is Mij. Here a11 = 1. So M11 = Minor of a11 = 3 M12 = Minor of the element a12 = 4 M21 = Minor of the element a21 = –2 M22 = Minor of the element a22 = 1

Now, cofactor of aij is Aij. So, 1+1 2 A11 = (–1) , M11 = (–1) (3) = 3 1+2 3 A12 = (–1) , M12 = (–1) (4) = –4 2+1 3 A21 = (–1) , M21 = (–1) (–2) = 2 2+2 4 A22 = (–1) , M22 = (–1) (1) = 1