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Unformatted text preview: 11/12/2020 Introduction to Matrices | Boundless Algebra Boundless Algebra Matrices Introduction to Matrices 1/15 11/12/2020 Introduction to Matrices | Boundless Algebra Ad Learn more What is a Matrix? A matrix is a rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. LEARNING OBJECTIVES Describe the parts of a matrix and what they represent KEY TAKEAWAYS Key Points A matrix (whose plural is matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. A matrix with m rows and n columns is called an m × n matrix or m-by-n matrix, where m and n are called the matrix dimensions. 11/12/2020 Introduction to Matrices | Boundless Algebra Matrices can be used to compactly write and work with multiple linear equations, that is, a system of linear equations. Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. Key Terms element: An individual item in a matrix row vector: A matrix with a single row column vector: A matrix with a single column square matrix: A matrix which has the same number of rows and columns matrix: A rectangular array of numbers, symbols, or expressions, arranged in rows and columns History of the Matrix The matrix has a long history of application in solving linear equations. They were known as arrays until the 1800‘s. The term “matrix” (Latin for “womb”, derived from mater—mother) was coined by James Joseph Sylvester in 1850, who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that are derived from the original one by removing columns and rows. An English mathematician named Cullis was the rst to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the rst signi cant use of the notation A = ai,j to represent a matrix where a i,j refers to the element found in the ith row and the jth column. Matrices can be used to compactly write and work with multiple linear equations, referred to as a system of linear equations, simultaneously. Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. What is a Matrix In mathematics, a matrix (plural matrices) is a rectangular array of 11/12/2020 Introduction to Matrices | Boundless Algebra numbers, symbols, or expressions, arranged in rows and columns. Matrices are commonly written in box brackets. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively. The size of a matrix is de ned by the number of rows and columns that it contains. A matrix with m rows and n columns is called an m × n matrix or m-by-n matrix, while m and n are called its dimensions.The dimensions of the following matrix are 2 × 3 up(read “two by three”), because there are two rows and three columns. A = [ 1 9 −13 20 5 −6 Matrix Dimensions: Each element of a matrix is often denoted by a variable with two subscripts. For instance, a represents the element at the second row and rst column of a matrix A. 2,1 The individual items (numbers, symbols or expressions) in a matrix are called its elements or entries. Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the rst equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated eld. 11/12/2020 Introduction to Matrices | Boundless Algebra Matrices which have a single row are called row vectors, and those which have a single column are called column vectors. A matrix which has the same number of rows and columns is called a square matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix. Addition and Subtraction; Scalar Multiplication Matrix addition, subtraction, and scalar multiplication are types of operations that can be applied to modify matrices. LEARNING OBJECTIVES Practice adding and subtracting matrices, as well as multiplying matrices by scalar numbers KEY TAKEAWAYS Key Points When performing addition, add each element in the rst matrix to the corresponding element in the second matrix. When performing subtraction, subtract each element in the second matrix from the corresponding element in the rst matrix. Addition and subtraction require that the matrices be the same dimensions. The resultant matrix is also of the same dimension. Scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. 11/12/2020 Introduction to Matrices | Boundless Algebra Key Terms scalar: A quantity that has magnitude but not direction. There are a number of operations that can be applied to modify matrices, such as matrix addition, subtraction, and scalar multiplication. These form the basic techniques to work with matrices. These techniques can be used in calculating sums, di erences and products of information such as sodas that come in three di erent avors: apple, orange, and strawberry and two di erent packaging: bottle and can. Two tables summarizing the total sales between last month and this month are written to illustrate the amounts. Matrix addition, subtraction and scalar multiplication can be used to nd such things as: the sales of last month and the sales of this month, the average sales for each avor and packaging of soda in the 2-month period. Adding and Subtracting Matrices We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries. In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. Matrix addition is commutative and is also associative, so the following is true: A + B = B + A (A + B) + C = A + (B + C) Adding matrices is very simple. Just add each element in the rst matrix to the corresponding element in the second matrix. 11/12/2020 Introduction to Matrices | Boundless Algebra ( 1 2 3 4 5 6 ) + ( 10 20 30 40 50 60 ) = ( 11 22 33 44 55 66 ) Note that element in the rst matrix, 1, adds to element x second matrix, 10, to produce element x 11 11 in the in the resultant matrix, 11. Also note that both matrices being added are 2 × 3, and the resulting matrix is also 2 × 3. You cannot add two matrices that have di erent dimensions. As you might guess, subtracting works much the same way except that you subtract instead of adding. 10 −20 30 40 50 60 ( 1 −2 3 4 −5 6 ) − ( 9 −18 27 36 55 54 ) = ( ) Once again, note that the resulting matrix has the same dimensions as the originals, and that you cannot subtract two matrices that have di erent dimensions. Be careful when subtracting with signed numbers. Scalar Multiplication In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. What does it mean to multiply a number by 3? It means you add the number to itself 3 times. Multiplying a matrix by 3 means the same thing; you add the matrix to itself 3 times, or simply multiply each element by that constant. 3 ⋅ ( 1 2 3 4 5 6 ) = ( 3 6 9 12 15 18 ) The resulting matrix has the same dimensions as the original. Scalar multiplication has the following properties: Left and right distributivity: (c + d)M = M(c + d) = Mc + Md Associativity: (cd)M = c(dM) Identity: 1M = M 11/12/2020 Introduction to Matrices | Boundless Algebra Null: 0M = 0 Additive inverse: (−1)M = −M Matrix Multiplication When multiplying matrices, the elements of the rows in the rst matrix are multiplied with corresponding columns in the second matrix. LEARNING OBJECTIVES Practice multiplying matrices and identify matrices that can be multiplied together KEY TAKEAWAYS Key Points If A is an n × m matrix and B is an m × p matrix, the result AB of their multiplication is an n × p matrix de ned only if the number of columns m in A is equal to the number of rows m in B. The product of a square matrix multiplied by a column matrix arises naturally in linear algebra for solving linear equations and representing linear transformations. Key Terms matrix: A rectangular arrangement of numbers or terms having various uses such as transforming coordinates in geometry, solving systems of linear equations in linear algebra and representing graphs in graph theory. 11/12/2020 Introduction to Matrices | Boundless Algebra If A is an n × m matrix and B is an m × p matrix, the result AB of their multiplication is an n × p matrix de ned only if the number of columns m in A is equal to the number of rows m in B. Check to make sure that this is true before multiplying the matrices, since there is “no solution” otherwise. General De nition and Process: Matrix Multiplication Scalar multiplication is simply multiplying a value through all the elements of a matrix, whereas matrix multiplication is multiplying every element of each row of the rst matrix times every element of each column in the second matrix. Scalar multiplication is much more simple than matrix multiplication; however, a pattern does exist. When multiplying matrices, the elements of the rows in the rst matrix are multiplied with corresponding columns in the second matrix. Each entry of the resultant matrix is computed one at a time. For two matrices the nal position of the product is shown below: ⎡ a11 a12 ⋅ ⋅ ⎢ ⎢ ⎢ a31 ⎣ ⋅ a32 ⋅ ⎤ ⎡ ⋅ ⎥ ⎥[ ⎥ ⋅ ⎦ b12 b13 b22 b23 ⋅ ⎢⋅ = ⎢ ⎢⋅ ⎣ ⋅ x12 ⋅ ⋅ ⋅ ⋅ ⋅ ⎤ ⎥ ⎥ x33 ⎥ ⋅ ⎦ 11/12/2020 Introduction to Matrices | Boundless Algebra B b1,1 b1,2 b1,3 b2,1 b2,2 b2,3 a1,1 a1,2 a2,1 a2,2 A a3,1 a3,2 a4,1 a4,2 Matrix Multiplication: This gure illustrates diagrammatically the product of two matrices A and B, showing how each intersection in the product matrix corresponds to a row of A and a column of B. The values at the intersections marked with circles are: x12 = (a11 , a12 ) ⋅ (b12 , b22 ) = (a11 b12 ) + (a12 b22 ) x33 = (a31 , a32 ) ⋅ (b13 , b23 ) = (a31 b13 ) + (a32 b23 ) Matrix Multiplication: Process Example 1: Find the product AB 1 2 3 4 A = ( ) 5 6 7 8 B = ( ) First ask: Do the number of columns in A equal the number of rows in B ? The number of columns in A is 2, and the number of rows in B is also 2 , therefore a product exists. 11/12/2020 Introduction to Matrices | Boundless Algebra Start with producing the product for the rst row, rst column element. Take the rst row of Matrix A and multiply by the rst column of Matrix B : The rst element of A times the rst column element of B, plus the second element of A times the second column element of B. (1 ⋅ 5) + (2 ⋅ 7) () + () () + () () + () AB = ( ) Continue the pattern with the rst row of A by the second column of B, and then repeat with the second row of A. AB has entries de ned by the equation: (1 ⋅ 5) + (2 ⋅ 7) (1 ⋅ 6) + (2 ⋅ 8) (3 ⋅ 5) + (4 ⋅ 7) (3 ⋅ 6) + (4 ⋅ 8) AB = ( ) (5 + 14) (6 + 16) (15 + 28) (18 + 32) AB = ( ) (19) (22) (43) (50) AB = ( ) The Identity Matrix The identity matrix [I ] is de ned so that [A][I ] = [I ][A] = [A], i.e. it is the matrix version of multiplying a number by one. LEARNING OBJECTIVES Discuss the properties of the identity matrix KEY TAKEAWAYS 11/12/2020 Introduction to Matrices | Boundless Algebra Key Points For any square matrix, its identity matrix is a diagonal stretch of 1s going from the upper-left-hand corner to the lower-right, with all other elements being 0. Non-square matrices do not have an identity. That is, for a non-square matrix [A], there is no matrix such that [A][I ] = [I ][A] = [A] . Proving that the identity matrix functions as desired requires the use of matrix multiplication. Key Terms matrix: A rectangular arrangement of numbers or terms having various uses such as transforming coordinates in geometry, solving systems of linear equations in linear algebra and representing graphs in graph theory. identity matrix: A diagonal matrix all of the diagonal elements of which are equal to 1, the rest being equal to 0. The number 1 has a special property: when multiplying any number by 1, the result is the same number, i.e. 5 ⋅ 1 = 5. This idea can be expressed with the following property as an algebraic generalization: 1x = x. The matrix that has this property is referred to as the identity matrix. De nition of the Identity Matrix The identity matrix, designated as [I ], is de ned by the property: [A][I ] = [I ][A] = [A] . Note that the de nition of [I][I] stipulates that the multiplication must commute, that is, it must yield the same answer no matter in which order multiplication is done. This stipulation is important because, for most matrices, multiplication 11/12/2020 Introduction to Matrices | Boundless Algebra does not commute. What matrix has this property? A rst guess might be a matrix full of 1s, but that does not work: ( 1 2 3 4 So ( )( 1 1 1 1 1 1 1 1 ) ) = ( 3 3 7 7 ) is not an identity matrix. The matrix that does work is a diagonal stretch of 1s, with all other elements being 0. ( 1 3 2 4 So ( )( 1 0 0 1 1 0 0 1 ) ) = ( 1 3 2 4 ) is the identity matrix for 2 × 2 matrices. For a 3 × 3 matrix, the identity matrix is a 3 × 3 matrix with diagonal 1s and the rest equal to 0: ⎛ 2 ⎜5 ⎝ π −3 1 −2 9 2 8 8.3 0 0 ⎟⎜0 1 0⎟ = ⎜5 0 1 ⎠⎝ 1 0 0 So ⎜ 0 1 0⎟ 0 1 ⎛ ⎝ 0 2 ⎞⎛1 ⎞ 0 ⎞ ⎠ ⎛ ⎝ 9 π −3 1 −2 8 2 8.3 ⎞ ⎟ ⎠ is the identity matrix for 3 × 3 matrices. ⎠ It is important to con rm those multiplications, and also con rm that they work in reverse order (as the de nition requires). There is no identity for a non-square matrix because of the requirement of matrices being commutative. For a non-square matrix [A] one might be able to nd a matrix [I ] such that [A][I ] = [A], however, if the order 11/12/2020 Introduction to Matrices | Boundless Algebra is reversed then an illegal multiplication will be left. The reason for this is because, for two matrices to be multiplied together, the rst matrix must have the same number of columns as the second has rows. ...
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