Section 3.4: Matrix Multiplication . If A is a matrix of size m x n and B is a matrix of size n x p then the product AB is defined and is a matrix of size m x p. So, two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. Example 1: Multiple the given matrices ...
multiply these matrices together. Example. Find 3 2 1 4! x y!. Solution.The first matrix has size 2 × 2. The second matrix has size 2 × 1. Clearly the number of columns in the first is the same as the number of rows in the second. The multiplication can be performed and the result will be a 2× 1 matrix. 3 2 1 4! x y! = 3x+2y x+4y! There ...
Page 1 of 2 208 Chapter 4 Matrices and Determinants Multiplying Matrices MULTIPLYING TWO MATRICES The product of two matrices A and B is defined provided the number of columns in A is equal to the number of rows in B. If A is an m ª n matrix and B is an n ª p matrix, then the product AB is an m ª p matrix. Describing Matrix Products
Multiplying matrices If A = 4 −13 1 −29 and B = 0 −5 −1 −4 then AB is not defined as A is a 2×3 matrix and B is a 2×2 matrix; the number of columns of A does not equal the number of rows of B. On the other hand, the product BA is defined as the number of columns of B, 2, does equal the number of rows of A. This tells us something very important; order matters!!
Matrix Multiplication Starter 1. For the matrix state the number that corresponds to these elements: (a) (b) (c) Notes When multiplying matrices the rows of the 1st matrix get multiplied by the columns of the 2nd matrix. E.g. Find . Working: Let be the result of the product of two matrices. is the element in the 1st row, 3rd column
1.4. Matrix Multiplication AB and CR 27 1.4 Matrix Multiplication AB and CR 1 To multiply AB we need row length for A = column length for B. 2 The number in row i, column j of AB is (row i of A) ·(column j of B). 3 By columns: A times column j of B produces column j of AB. 4 Usually AB is differentfrom BA. But always (AB)C = A(BC). 5 If A has r independentcolumnsin C, then A = CR = (m×r)(r ×n).
M3 Matrix Multiplication Matrices may be added and subtracted if they have the same shape. That is, the number of rows and columns is the same. Matrices may also be multiplied. However the requirements for multiplication are very different to that for addition/subtraction. This module looks at matrix multiplication. Matrix Shape or Order of a ...
Matrix algebra: linear operations Addition: two matrices of the same dimensions can be added by adding their corresponding entries. Scalar multiplication: to multiply a matrix A by a scalar r, one multiplies each entry of A by r. Zero matrix O: all entries are zeros. Negative: −A is defined as (−1)A. Subtraction: A−B is defined as A+(−B).
First consider the size of each of these matrices: A is a 2× 2 matrix, B is 3× 2, C is 2× 3 and D is 3× 2. From these four matrices only B and D are compatible. This means we can calculate B+D, B−D, D −B, but we cannot add or subtract any other pair of these matrices. The matrix CT = 3 0 0 2 −3 1
matrix operations. Addition and Scalar Multiplication: Just like with vector operations, the sum of matrices and the multiplica-tion by a scalar (just a number, as opposed to a vector or matrix) are done component-by-component. 1 Try the following matrix operations: (a)3 " 1 0 0 1 # (b) " 1 1 1 1 # + " 0 1 1 0 # (c) " 1 2 2 4 # " 2 1 3 0 # (d ...
2 Matrix-vector multiplication Row-sweep algorithm Column-sweep algorithm 3 Matrix-matrix multiplication \Standard" algorithm ijk-forms CPS343 (Parallel and HPC) Matrix Multiplication Spring 2020 14/32. Using a two-dimensional arrays It is natural to use a 2D array to store a dense or banded matrix.
Theorem6.2.2: Matrix Multiplication, Linear Combination of Columns Method Let A and B be m×n and n×p matrices respectively. The product C= AB is the matrix for which c∗j = b1ja∗1 +b2ja∗2 +b3ja∗3 +···+b nja∗n That is, the jth column c∗j of C is the linear combination of all the columns of A,
2.10 Example Commutativity can fail more dramatically: 56 78! 12 0 34 0! = 23 34 0 31 46 0! while 12 0 34 0! 56 78! isn’t even defined. 2.11 Remark The fact that matrix multiplication is not commutative can seem odd at first, perhaps because most mathematical operations in prior courses are commutative. But matrix multiplication represents ...
Matrix Multiplication worksheet MATH 1010/1210/1300/1310 Instructions: Perform each multiplication below, or state why it can’t be done. 1. 2 6 6 6 6 4 5 ...
2. More general matrix multiplication When we multiplied matrices in the previous section the answers were always single numbers. Usually however, the result of multiplying two matrices is another matrix. Two matrices can only be multiplied together if the number of columns in the first matrix is the same as the number of rows in the second.
3.2. Multiplication of Matrices and Multiplication of Vectors and Matrices 3 Definition. For a given matrix A, we may perform the following operations: Row Interchange: Form matrix B by interchanging row i and row j of matrix A, denoted A R i↔R j] B. Row Scaling: Form matrix B by multiplying the ith row of A be a nonzero scalar s, denoted A ...
2 3 Matrix Multiplication Suppose we buy two CDs at $3 each and four Zip disks at $5 each. We calculate our total cost by computing the products’ price ××× quantity and adding: Cost = 3 ××× 2 + 5 ×××× 4 = $26 . Let us instead put the prices in a row vector P = [3 5] and the quantities purchased in a column vector, The price matrix
Strassen’s algorithm can be used to multiply two 2n 2n matrices. The key is to write the matrices in block form: A 11 A 12 A 21 A 22 B 11 B 12 B 21 B 22 = C 11 C 12 C 21 C 22 Each of the blocks is a 2n 1 2n 1 matrix. Applying Strassen’s algorithm to the big matrices, there are some additions and subtractions that we already know how to do ...
“main” 2007/2/16 page 123 2.2 Matrix Algebra 123 then Ac = c1a1 +c2a2 = 5 2 4 +(−1) −1 3 = 11 17 . Case 3: Product of an m×n matrix and an n×p matrix. If A is an m×n matrix and B is an n × p matrix, then the product AB has columns defined by multiplying the matrix A by the respective column vectors of B, as described in Case 2.That is, if B =[b1,b2,...,bp], then AB is the m×p ...
