Prepared by Deluar Jahan Moloy Lecturer Northern University Bangladesh Matrix Algebra Prepared by Deluar Jahan Moloy Lecturer Northern University Bangladesh

Matrix A matrix is a rectangular or square array of numbers or values arranged in rows and columns enclosed by a pair of brackets or a pair of double bars or a pair of parentheses. Matrix are generally denoted by capital letters such as A, B, C….. For examples,

Rows and Columns The horizontal and vertical lines in a matrix are called rows and columns respectively. For example Row column

Elements The numbers or values in the matrix are called its entries or its elements. For example, Here 10, 5, 3, 2, 6 and 7 are called elements.

Dimension or its order The number of rows and columns that a matrix has is called its dimension or its order. It is denoted by .Where m represents no. of rows and n represents no. of columns. By convention, rows are listed first and columns are second.

Examples We would say that the dimension (or order) of the 3rd matrix is 2 x 3, meaning that it has 2 rows and 3 columns.

Row matrix A matrix with only one row (a 1 × n matrix) is called a row matrix. For example, A = is a 1 × 4 matrix.

Column matrix A = is a 4 × 1 matrix. A matrix with one column (an m × 1 matrix) is called a column matrix. For example, A = is a 4 × 1 matrix.

Square matrix is a 2× 2 square matrix. is a 3 × 3 square matrix. A matrix is said to be square matrix if no. of rows and no. of columns are equal. For example, is a 2× 2 square matrix. is a 3 × 3 square matrix.

Unit or Identity matrix A square matrix whose diagonal elements are unity (or one) and the remaining elements are zero is called Identity matrix. For example

Null or Zero Matrix A matrix each of whose elements are zero is called zero or null matrix. For example, are null or zero matrices.

Transpose Matrix The transpose of one matrix is another matrix that is obtained by using rows of the first matrix as columns of second matrix. For example, Then transpose of a is denoted by

Equal matrices Two matrices are said to be equal if and only if 1. They are of the same order, i.e. they have the same number of rows and columns and 2. Each element of one is equal to the corresponding elements of the other.

Equal matrix For example, and are equal matrices.

Exercise 1 Given A= B= If A=B, find the value of a, b, c, d and e.

Exercise 2 Find the value of x and y if

Addition of matrices To add two matrices A and B: # of rows in A = # of rows in B # of columns in A = # of columns in B The sum is obtained by adding each element of A with corresponding element of B.

Subtraction of matrices To subtract two matrices A and B: # of rows in A = # of rows in B # of columns in A = # of columns in B The difference is obtained by subtracting each element of B from the corresponding element of A.

Addition and subtraction of matrices If Then A + B = possible or not C + B = possible or not A + C = possible or not A - B = possible or not C - B = possible or not A - C = possible or not

Addition and subtraction of matrices Form previous, A + C = + = =

Addition and subtraction of matrices Form previous, A - C = - = =

Addition

Exercise1 If A= B= Find i) A+B ii) A-B

Scalar Multiplication of a matrix A real number is referred to as a scalar when it occurs in operations involving matrices. If A is an m × n matrix and k is a scalar, then we let kA denote the matrix obtained by multiplying every element of A by k.

Examples If A= Find 2A 2A=2.A=2. =

Exercise1 If A= B= Find the value of 5A+3B

Multiplication of Matrices If A and B are two matrices such that the number of columns (n) in A is equal to the number of rows (m) in B, then the product of A and B denoted by AB. (The number of columns (n) in first matrix must equal the number of rows (m) in 2nd matrix in order to carry out the matrix multiplication.)

Multiplication of Matrices To multiply two matrices A and B # of columns in A (1st matrix) = # of rows in B (2nd matrix) If and . Then matrix multiplication is possible. The product is denoted by AB=A.B.

Example1 If B= C= BC=B.C=

Example2 If A= , find AB BA. Is AB = BA?

Example 3 If A= Find AB and BA if they are possible.

Example4 Given that Find