Matrices: 2.1 Operations with Matrices

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2 Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices, to be called matrix multiplication.

3 Definition Preview Matrices were defined in 1.2 as an array, with m rows and n columns: a 11 a 12 a 13 a 1n a 11 a 12 a 13 a 1n a 21 a 22 a 13 a 2n a 31 a 32 a 33 a 3n Or a 21 a 22 a 13 a 2n a 31 a 32 a 33 a 3n a m1 a m2 a m3 a mn a m1 a m2 a m3 a mn where a ij are real numbers (for this class).

4 Size, row and column matrix This matrix above is said to have size m n, because it has m rows and n columns. A matrix with equal number of rows and columns is called a square matrix. A square matrix of size n n is said to have order n. If a matrix has only one row, it is called a row matrix. Likewise, if a matrix has only one column, it is called a column matrix.

5 Notations 1. Often, a matrix is denoted by uppercase letters: A, B, We also denote the above matrix as [a ij ]. 3. We may write A = [a ij ].

6 Equality Preview Two matrices A = [a ij ], B = [b ij ] are equal, if they have same size (m n) and a ij = b ij for 1 i m, 1 j n.

7 Addition Preview If A = [a ij ], B = [b ij ] are two matrices of same size m n, then their sum is defined to be the m n matrix given by A + B = [a ij + b ij ]. So, the sum is obtained by adding the respective entries. If the sizes of two matrices are different, then the sum is NOT defined.

8 In this context of matrices of real numbers, by a scalar we mean a real number. If A = [a ij ] is a m n matrix and c is a scalar, then the scalar multiplication of A by c is the m n matrix given by ca = [ca ij ]. Therefore, ca is obtained by multiplying each entry of A by c.

9 of Addition Let A = 3 10, B = 7 a 7 3 b Then A + B = a, 7 + b 0

10 of scalar multiplication Let A = [ Then scalar multiplication by 11 gives [ ] A = ]

11 Suppose A = [a ij ] is a matrix of size m n and B = [b ij ] is a matrix of size n p. Then, the product AB is an m p matrix n AB = [c ij ] where c ij = a ik b kj = a i1 b 1j +a i2 b 2j + +a in b nj. k=1 Remarks. The (i, j) th entry c ij is obtained by combining the i th row of A and j th column of B. We required that the number of columns of A is equal to the number of rows of B. If they are unequal, then the product AB is NOT defined.

12 = a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn c 11 c 12 c 1p c 21 c 22 c 2p c m1 c m2 c mp b 11 b 12 b 1p b 21 b 22 b 2p b n1 b n2 b np c 12 = a 11 b 12 +a 12 b a 1n b n2

13 of matrix multiplication Let A = [ ], B = Since number of columns of A and number of rows of B are same, the product AB is defined.,

14 We have [ ] ( 3) ( 3) 1 AB = ( 3) ( 3) 1 = [ Remark. Note BA is ALSO defined, which will be a 3 3 matrix. You can compute it similarly. ]

15 Matrix form of Linear Systems A system of linear linear equations can be written in a matrix form: Ax = b where a x 1 b 11 a 12 a 13 a 1n 1 x = x 2, b = b 2, A = a 21 a 22 a 13 a 2n a 31 a 32 a 33 a 3n x n b m a m1 a m2 a m3 a mn Here A is the coefficient matrix.

16 2.1.1 Preview The solutions of a system can also be written in the matrix form. The system of equations { 2x y z = 0 is same as x +3y z = 0 ( ) x y z = ( 0 0 )

17 Continued Preview Its solutions, can be (computed and) written in one of the two ways: x 4 x = 4t, y = t, z = 7t OR y z = t 1 7 where t is a parameter.

18 2.1.2 Preview The system from 1.1 x 1 +4x 3 = 13 2x 1 x 2 +.5x 3 = 3.5 2x 1 2x 2 7x 3 = x 1 x 2 x 3 = is same as

19 Continued Preview Its solution, computed in 1.2 can be written as x x 2 = t 7.5 x 3 0 1

20 2.1.3 Preview The system from 1.1 x 1 +3x 4 = 4 6x 2 3x 3 3x 4 = 0 3x 2 2x 4 = 1 2x 1 x 2 +4x 3 = x 1 x 2 x 3 x 4 = is same as

21 Continued Preview Its solution can be written as x 1 x 2 x 3 x 4 =