d i E Matrices a l l u d Dr. Abdulla Eid b A College of - PowerPoint PPT Presentation

Section 6.1 d i E Matrices a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 103: Mathematics for Business I Dr. Abdulla Eid (University of Bahrain) Matrices 1 / 11

Goal d We want to learn i E 1 What a matrix is? a 2 How to add or subtract two matrices? l l u 3 How to multiple two matrices? d b 4 How to find the multiplicative inverse? A 5 What is the determinant of a matrix and why it is useful? . r D 6 How to solve system of linear equations using matrices? Dr. Abdulla Eid (University of Bahrain) Matrices 2 / 11

1- Matrices Definition A matrix is just a rectangular array of entries. It is described by the rows and columns . d i note: The work matrix is singular. The plural of matrix is matrices E (pronounced as‘may tri sees‘). a l l u Example d b � 2 � � 5 � 0 6 1 � � A A = B = C = 1 2 3 3 2 7 1 2 . r D Usually the matrices are written in the form � B 11 � B 12 B 13 , with B i j B 21 B 22 B 23 ���� ���� row column Dr. Abdulla Eid (University of Bahrain) Matrices 3 / 11

Definition An m × n –matrix is a rectangular array consists of m rows and n columns. d i  E  · · · A 11 A 12 A 1 n a A 21 A 22 · · · A 2 n   l   l · · · · u   A = = ( A ij ) m × n  d  · · · ·   b   · · A · ·   · · · A n 1 A n 2 A nn . r D where A ij is the entry in the row i and column j . Dr. Abdulla Eid (University of Bahrain) Matrices 4 / 11

Example Let d   3 − 2 7 3 i E − 1 − 5 2 1   A =  a  4 3 2 1   l l u 0 8 0 2 d b A 1 What is the size of A ? . 2 Find A 21 , A 42 , A 32 , A 34 , A 44 , A 55 . r D 3 What are the entries of the second row? Dr. Abdulla Eid (University of Bahrain) Matrices 5 / 11

Definition If A is a matrix, the transpose of A is a new matrix A T formed by interchanging the rows and the columns of A , i.e., A T = ( A ji ) d i E Example a l Find the transpose M T and ( M T ) T . l u d b � 6 � � 2 � − 3 1 3 � � A A = B = C = 3 1 2 5 2 4 7 1 6 . r D Solution: 1 � 6 � � 6 � 2 − 3 A T = ( A T ) T = and − 3 4 2 4 Dr. Abdulla Eid (University of Bahrain) Matrices 6 / 11

Example Find the transpose M T and ( M T ) T . � 6 � � 2 � − 3 1 3 � � A = B = C = 3 1 2 5 2 4 7 1 6 d i E a Solution: l l u 1 d   2 7 b � 5 � 6 1 B T = ( B T ) T = A 1 1 and   7 1 2 3 6 . r D 2   3 1 C T =   ( C T ) T = � � and 3 1 2 5   2   5 Dr. Abdulla Eid (University of Bahrain) Matrices 7 / 11

Note: 1 ( A T ) T = A . 2 A matrix A is called symmetric if A T = A . Question: When two matrices are equal? Definition d Two matrices A and B are equal if they have the same size and the same i E entries at the same position, i.e., a l l u A ij = B ij d b A Example . Solve the matrix equation r D     4 2 1 4 2 1  = 2 y 3 z − 3 − 8 0 x    0 1 2 0 1 2 Solution: x = − 3, 2 y = − 8 → y = − 4, 3 z = 0 → z = 0 Dr. Abdulla Eid (University of Bahrain) Matrices 8 / 11 Solution Set = { ( − 3, − 4, 0 ) }

Special Matrices Zero matrix 0 m × n = ( 0 ) m × n “zero everywhere“.   0 0 0 � 0 � � 0 � 0 0 � �  , 0 0 , , 0 0 0  0 0 0 0 d 0 0 0 i E Square matrix if m = n (having the same number of rows and a l columns). l u d   6 5 − 1 � 3 b � 2 � � A 3 , , 1 3 3   1 − 5 8 − 9 0 . r D Diagonal matrix if it is a square matrix ( m = n ) and all entries off the main diagonal are zeros.   3 0 0 0   6 0 0 � 3 � 0 0 − 5 0 0    , , 0 3 0    − 5 − 11 0 0 0 0   0 0 6 0 0 0 2 Dr. Abdulla Eid (University of Bahrain) Matrices 9 / 11

Special Matrices Upper Diagonal matrix if it has zeros below the main diagonal (entries are ‘upper‘ the main diagonal).   d − 7 3 1 2   6 3 5 i � 3 � E 5 0 − 5 − 4 6    , , 0 3 − 2    − 5 − 11 0 a 0 0 6   0 0 6 l l 0 0 0 2 u d b A Lower Diagonal matrix if it has zeros above the main diagonal (entries are ‘lower‘ the main diagonal). . r D   3 0 0 0   6 0 0 � 3 � 0 6 − 5 0 0    , , 3 3 0    − 5 − 8 − 4 − 11 7 0   4 7 6 1 4 7 2 Dr. Abdulla Eid (University of Bahrain) Matrices 10 / 11

Special Matrices Row vector is a matirx with only one row. � � � � � � � � − 2 2 3 , 5 13 12 , 7 3 0 6 , 0 0 0 d i Column vector is a matrix with only one column. E   a 6   3 l � 3 � l 1 u    , , 1    d 6 8   b − 5 0 A . Identity matrix I n if m = n and has one in the main diagonal and zero r D elsewhere.   1 0 0 0   1 0 0 � 1 � 0 0 1 0 0   � �  , I 1 = I 2 = I 3 = I 4 = 1 , , 0 1 0    0 1 0 0 1 0   0 0 1 0 0 0 1 Dr. Abdulla Eid (University of Bahrain) Matrices 11 / 11