Matrix Addition and Subtraction
A matrix is a rectangular array of numbers, enclosed by brackets. The numbers in a matrix are arranged in horizontal rows and vertical columns. Each number in a matrix is an entry. The dimensions of a matrix are its numbers of rows and columns, written in the form , where is the number of rows and is the number of columns.
Performing matrix calculations involves examining each entry in a row or column, locating entries, and writing and reading the notation for an entry. A general notation of can represent any entry in a matrix, with as the row number and as the column number.For example, matrix is a matrix because it has 3 rows and 4 columns.
To add or subtract matrices, add or subtract corresponding entries. The matrices must have the same dimensions so that every entry in one matrix has a corresponding entry in the other matrix.For example, matrix and matrix have the same dimensions. They are each a matrix.
A scalar is a real number or a quantity that can be represented by a real number. To determine the product of a scalar and a matrix, multiply each entry in the matrix by the scalar. The result is called a scalar product.For example, matrix can be multiplied by 4 to determine matrix . Multiply the entry in row 1 and column 1 of matrix by the scalar, which is 4. Then, place the product in row 1 and column 1 of matrix . Continue multiplying each entry by the scalar until matrix is complete.
Matrix Product Dimensions
To multiply two matrices, multiply consecutive entries in row of the first matrix and column in the second matrix. The sum of those products will become the entry in row and column in the product matrix.
Unlike multiplication with real numbers, matrix multiplication is not commutative, so may not be equal to , even if both matrices are defined.
The numbers 2 and are multiplicative inverses because when multiplied, the result is the multiplicative identity 1. Square matrices may also have multiplicative inverses. A square matrix is a matrix that has the same number of rows and columns.An identity matrix is a square matrix in which the entries (or numbers) along the main diagonal, from the top left corner to the bottom right corner of the matrix, are ones and the other entries are zeros. The identity matrix has an entry of 1 in positions , , and . All other entries are zero.
Sometimes, it is useful to determine the inverse of a matrix. This is because there is no such thing as matrix division. With numbers, it is possible to multiply by the reciprocal of the number and get the same result as dividing by that number. Similarly, multiplying by the inverse of a matrix is equivalent to dividing by the matrix.