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.
The resulting matrix has the same dimensions as the original. Scalar multiplication has the following properties:. When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix. 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 first 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. Each entry of the resultant matrix is computed one at a time. Matrix Multiplication: This figure 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. Start with producing the product for the first row, first column element. The matrix that has this property is referred to as the identity matrix.
Note that the definition 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.
What matrix has this property? It is important to confirm those multiplications, and also confirm that they work in reverse order as the definition requires. There is no identity for a non-square matrix because of the requirement of matrices being commutative. The reason for this is because, for two matrices to be multiplied together, the first matrix must have the same number of columns as the second has rows.
Privacy Policy. Skip to main content. Search for:. Introduction to Matrices. 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.
Matrices can be used to compactly write and work with multiple linear equations, that is, a system of linear equations. 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.
If you like what you see, we hope you will consider buying. Get the App. Algebra More Algebra. Who invented matrices? It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations. The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive. For example a tablet dating from around BC contains the following problem:- There are two fields whose total area is square yards.
If the total yield is bushels, what is the size of each field. Indeed it is fair to say that the text Nine Chapters on the Mathematical Art written during the Han Dynasty gives the first known example of matrix methods. First a problem is set up which is similar to the Babylonian example given above:- There are three types of corn, of which three bundles of the first, two of the second, and one of the third make 39 measures. Two of the first, three of the second and one of the third make 34 measures.
And one of the first, two of the second and three of the third make 26 measures. How many measures of corn are contained of one bundle of each type? Now the author does something quite remarkable.
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