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.
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).
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.
Notations 1. Often, a matrix is denoted by uppercase letters: A, B,.... 2. We also denote the above matrix as [a ij ]. 3. We may write A = [a ij ].
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.
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.
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.
of Addition 1 1 0 3 Let A = 3 10, B = 7 a 7 3 b 3 1 2 Then A + B = 10 10 + a, 7 + b 0
of scalar multiplication Let A = [ 1 1 3 10 7 3 Then scalar multiplication by 11 gives [ ] 11 11 33 11A = 110 77 33 ]
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.
= 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 22 + +a 1n b n2
of matrix multiplication Let A = [ 1 1 3 10 7 3 ], B = 1 1 1 0 2 1 Since number of columns of A and number of rows of B are same, the product AB is defined.,
We have [ ] 1 1 + 1 1 + ( 3) 2 1 1 + 1 0 + ( 3) 1 AB = 10 1 + 7 1 + ( 3) 2 10 1 + 7 0 + ( 3) 1 = [ 4 2 11 7 Remark. Note BA is ALSO defined, which will be a 3 3 matrix. You can compute it similarly. ]
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.
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 ( 2 1 1 1 3 1 ) x y z = ( 0 0 )
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.
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 = 19 1 0 4 2 1.5 2 2 7 x 1 x 2 x 3 = is same as 13 3.5 19
Continued Preview Its solution, computed in 1.2 can be written as x 1 13 4 x 2 = 22.5 + t 7.5 x 3 0 1
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 = 5 1 0 0 3 0 6 3 3 0 3 0 2 2 1 4 0 x 1 x 2 x 3 x 4 = is same as 4 0 1 5
Continued Preview Its solution can be written as x 1 x 2 x 3 x 4 = 1 1 1 1