{"id":6718,"date":"2014-09-24T16:35:13","date_gmt":"2014-09-24T16:35:13","guid":{"rendered":"https:\/\/schooltutoring.com\/scholarship\/?p=6718"},"modified":"2014-12-02T08:25:25","modified_gmt":"2014-12-02T08:25:25","slug":"math-review-of-multiplication-of-matrices","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/scholarship\/2014\/09\/24\/math-review-of-multiplication-of-matrices\/","title":{"rendered":"Math Review of Multiplication of Matrices"},"content":{"rendered":"

Overview:<\/span><\/h3>\n

The dimensions of a matrix are important to its definition.\u00a0 In order for the multiplication of two matrices to be meaningful, each matrix must have certain dimensions.\u00a0 Multiplication of matrices is not commutative.<\/p>\n

Review of Scalar Multiplication<\/h3>\n

In order to multiply a matrix by a constant, every member of the matrix is multiplied by the constant.\u00a0 Suppose that matrix A consists of the members\u00a0[wxyz].\u00a0 Multiplying each member by a constant e means that the new matrix will consist of [ew ex ey ez]. Suppose Matrix B consists of [9 3 7 6].\u00a0 Multiplying each member by 3 would lead to the new matrix\u00a0[27 9 21 18].<\/p>\n

Figure 1: Matrix A<\/p>\n

<\/a><\/p>\n

Figure 2: In Matrix 1A, every element of Matrix A is multiplied by the constant e.<\/p>\n

<\/a><\/p>\n

Figure 3: Matrix B is a 2 X 2 Matrix, in the same form as Matrix A, but the variables are replaced with real numbers.<\/p>\n

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Figure 4: In Matrix 1B, every element of Matrix B is multiplied by 3<\/span><\/p>\n

<\/a><\/p>\n

What Dimensions Must the Matrix Be?<\/h3>\n

The first step in multiplying matrices is to determine the dimensions of each matrix.\u00a0 If the number of columns of the first matrix is equal to the number of rows in the second matrix, multiplying the matrices is possible.\u00a0 Suppose Matrix D consists of the elements\u00a0[1 3 5 7]\u00a0\u00a0and Matrix E consists of the elements [ 2 4 6 8 10 12].\u00a0 Matrix D has the dimensions 2 X 2 and Matrix E has the dimensions 2 X 3.\u00a0 Matrix DE would also have the dimensions 2 X 3.\u00a0 It would be equal to multiplying the elements of Matrix D\u00a0by the elements of\u00a0Matrix E.<\/p>\n

Figure 5: Matrix D<\/p>\n

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Figure 6: Matrix E<\/p>\n

<\/a><\/p>\n

What Is the Next Step in Multiplying Matrices?<\/h3>\n

In the next step, multiply each element of the row in the first matrix by each element of the column of the second matrix, and add those products together to form the new elements of the product matrix.\u00a0 Using the example of Matrix DE, the first row of the new matrix would equal (1\u22192 + 3\u22198) (1\u22194 + 3\u221910) (1\u22196 + 3\u221912), or (2 + 24)\u00a0 (4 + 30) (6 + 36).\u00a0 The second row of the new matrix DE would equal each element of the second row of the first matrix multiplied by each element of the column of the second matrix or (5\u22192 + 7\u22198) (5\u22194 + 7\u221910) (5\u22196 + 7\u221912) or (10 + 56) (20 + 70) (30 + 84).\u00a0 The new matrix would consist of the elements [26 34 42 66 90 114] in a 2 x 3 matrix.\u00a0.<\/p>\n

Figure 7: Matrix DE<\/p>\n

<\/a><\/p>\n

Why Isn’t Multiplication of Matrices Commutative?<\/h3>\n

Suppose Matrix F consisted of the elements [1 2 3 4] in a 2 x 2 matrix\u00a0and Matrix G consisted of the elements [ 0 1 3 5 7 9] in a 2 x 3 matrix..\u00a0 The new Matrix FG would consist of the elements [10 15 21\u00a020 31 44] in a 2 x 3 matrix. Matrix GF would be a 2 X3 matrix X a 2 X2 matrix, which would be undefined, as the number of columns in G (3) is not equal to the number of rows in F(2).<\/p>\n

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Overview: The dimensions of a matrix are important to its definition.\u00a0 In order for the multiplication of two matrices to be meaningful, each matrix must have certain dimensions.\u00a0 Multiplication of matrices is not commutative. Review of Scalar Multiplication In order to multiply a matrix by a constant, every member of the matrix is multiplied by […]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[2608,2607],"class_list":["post-6718","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-dimensions","tag-multiplication-of-matrices"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/6718","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/comments?post=6718"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/posts\/6718\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/media?parent=6718"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/categories?post=6718"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/scholarship\/wp-json\/wp\/v2\/tags?post=6718"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}