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An Introduction to Matrix Algebra King Saud University Matrix Operations • Although we introduced matrices as a structure for convenient “bookkeeping” when solving systems of linear equations, they are interesting mathematically in their own right. • We can define the operations of addition, scalar multiplication, subtraction and multiplication on them. Matrix Terminology • We say that two mxn matrices A and B are equal if they have the same size and aij=bij for 1≤ i ≤ m and 1 ≤ j ≤ n. • A matrix with only one row is called a row matrix or row vector. • A matrix with only one column is called a column matrix and column vector. (ai) Matrix Addition • If A and B are 2 mxn matrices then A+B is the mxn matrix with entries (a+b)ij= aij+bij. Scalar Multiplication • If A is an mxn matrix and c is a scalar then cA is the mxn matrix with entries, (ca)ij = caij. • With this definition we can define A-B to be A+(-1)B. Matrix Multiplication • If A is an mxp matrix and B is an pxn matrix then AB is the mxn matrix with entries, p (ab )ij aik bkj . k 1 Why???? • Consider the system of equations, a1,1 x a1,2 y a1, 3 z b1 a2,1 x a2,2 y a2,3 z b2 a3,1 x a3,2 y a3,3 z b3 • If A is the coefficient matrix of this system x is the column matrix of variables and b is the column matrix of right hand side numbers then the system is expressed as Ax = b. Partitioned Matrices • Sometimes it is also useful to think of a linear system in the following way: a11 a12 a21 a22 x1 x2 am 1 am 2 a1n b1 a2 n b2 xn amn b m • We say here that we have partitioned A into its column matrices a1, a2, ... , an, and written b as a linear combination of a1, a2, ... , an.