Any matrix which has as many columns as rows is called a square matrix. An n x n diagonal matrix whose entries-from the upper‐left to the lower‐right-are a 11, a 22,…, a nnis often written Diag( a 11, a 22,…, a nn). Blocks of zeros are often left blank in nondiagonal matrices also. For example,Īre two ways of writing exactly the same matrix. It is not uncommon in such cases, particularly with large matrices, to simply leave blank any entry that equals zero. For example, the matrix 3x3 above is a diagonal matrix. If every off‐diagonal entry of a matrix equals zero, then the matrix is called a diagonal matrix. In the matrix A, the diagonal entries are a 11 = 2 and b 22 = 4 in B, the diagonal entries are b 11 = 2 and b 22 = 4 and in the matrix 3x3, the diagonal entries are δ 11 = δ 22 = δ 33 = 1. The diagonal entries in each of the following matrices are highlighted: Any entry whose column number matches its row number is called a diagonal entry all other entries are called off‐diagonal. Thus, the matrix isĮntries along the diagonal. The (1,1), (2,2), and (3,3) entries are each equal to 1, but all other entries are 0. Therefore,Įxample 3: Give the 3 x 3 matrix whose ( i, j) entry is expressed by the formula An example of such a matrix isĮxample 2: If B is the 2 x 2 matrix whose ( i, j) entry is given by the formula b ij= (−1) i+j( i+j), explicitly determine B. Since every matrix in M 2x3( R) consists of 2 rows and 3 columns, A will contain 2 x 3 = 6 entries. If A ε M 2x3( R), how many entries does the matrix A contain? Indicates that A is the m x n matrix whose ( i, j) entry is a ij.Įxample 1: The set of all m x n matrices whose entries are real numbers is denoted M m x n( R). In general, the ( i, j) entry of a matrix A is written a ij, and the statement The (1, 2) entry is 0, the (2, 3) entry is 1, and so forth. For example, since the entry −2 in the matrix above is in row 2, column 1, it is the (2, 1) entry. The entry in row i, column j is called the ( i, j) entry. The numbers in the array are called the entries of the matrix, and the location of a particular entry is specified by giving first the row and then the colun where it resides. Note that the rows are counted from top to bottom, and the columns are counted from left to right. For example, the matrices above are 2 by 3, since they contain 2 rows and 3 columns: If the matrix consists of m rows and n columns, it is said to be an mby n(written m x n) matrix. The size or dimensions of a matrix are specified by stating the number of rows and the number of columns it contains. A rectangular array of numbers, enclosed in a large pair of either parentheses or brackets, such as