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Lecture 12

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MATH 136
Robert Sproule

Friday, January 31 − Lecture 12 : Matrices and matrix algebra Concepts: 1. matrix and the operation of addition, multiplication, scalar multiplication and transposition on matrices. 2. rules of matrix algebra. 3. zero matrix 4. matrix-vector multiplication. 5. system of linear equations expressed as a matrix equation. 12.1 Definition − Matrices. If m and n are positive integers, a matrix of size m × n (or of dimension m × n), is a rectangular array of real numbers, arranged inm rows and n column. It is represented as A = [a ]ij m × n The a ’ijare called the entries orcomponents of the matrix. The symbol i is called the row index or the entrée and j is called its column index. Example : This is a 3 by 3 matrix A : 12.1.1 Two matrices are said to be equalmatrices if and only if their corresponding entries are equal. Because of this matrices of different dimensions cannot be equal. 12.1.2 Remark − A vector, such as u = (2, 1, 7) for example, can be also be expressed in matrix form, u = [2, 1, 7] 1 × 3, or The form it should take is usually determined by the context. 12.2 Definition − If A = [a ] ij m×n then the vectors r = ia , ai1 .i2, a ) iinℝ are called the n row-vectors of A (1 ≤ i ≤ m) . The vectors c j= (a 1ja 2j..., anj in ℝ are called the column-vectors of A (1 ≤ j ≤ n). So, when convenient, we can think of A as a pile of row- vectors, r , r , ..., r all stacked one on top the other, or as a juxtaposition of column 1 2 m vectors c ,1c 2 ..., cn: 12.3 Definitions − A matrix that contains only zeroes as entries is called the zero matrix. 12.4 Operations with matrices − Let M m×n denote the set of all m× n matrices with real numbers as entries. Then will define addition, scalar multiplicationand transposition on matrices in M m×n as follows: Operation What we start with : What we get : Addition Scalar multiplication Transposition Suppose then from the definition we see that the transpose of A exchange rows with columns: 12.4.1 Example – Let Aand B be the given matrices: T a) Compute 3A – 2B Answer: b) Find the (2, 3) entry of C = A + 3B. T Answer: Let C = [c ], ij= [b ], ij= A = [d ]. ij c23 d + 23 = a 233b =327 + 3(23 = −7 th T c) Find the (1, 3) entry of H = (2A + B) Answer: 4 th T d) Find the (2, 2) entry of E = 2A + A 12.4.2 Theorem – Let m and n be fixed positive integers. Then the set M m × nof all m × n matrices with real number entries equipped with + and scalar multiplication described above satisfiesthe following basic properties: 1) M is closed under addition and scalar multiplication. m × n 2) Addition in M is associative and commutative. m × n 3) Matrices in M distribute over finite sums of scalars. m × n 4) Scalars distribute over finite sums of matrices. 5) Transpose properties: i) (A ) = A, ii) (A + B) = A + B , T iii) (cA)T = cA T Proof : The proof of these follow immediately from the definition of + and scalar multiplication and transposition and so is omitted. 12.5 Some special matricesand matrix related definitions T  The square matrix A is said to be symmetric if A = A . That is, if A = [a ] ij n×nhen a ija .ji  A column matrix is a matrix which has only one column. While a row matrix is a matrix with only one row. Both a column matrix and row matrix can be referred to as vectors when useful and appropriate.  Given a square matrix A , tn×ntrace of the matrix
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