2 Chapter 1 A Brief Review of Matrices and Vectors denoted tr(A),is the sum of the elements in the main diagonal of A.Two matrices A and B are equal if and only if they have the same number of rows and columns,and aij =bij for all i and j. The transpose of an m x n matrix A,denote AT,is an n x m matrix obtained by interchanging the rows and columns of A.That is,the first row of A becomes the first column of AT,the second row of A becomes the second column of AT,and so on.A square matrix for which AT =A is said to be symmetric. Any matrix X for which XA I and AX I is called the imerse of A.Usually, the inverse of A is denoted A-1.Although numerous procedures exist for computing the inverse of a matrix,the procedure usually is to use a computer program for this purpose,so we will not dwell on this topic here.The interested reader can consult any book an matrix theory for extensive theoretical and practical discussions dealing with matrix inverses.A matrix that possesses an inverse in the sense just defined is called a nonsingular matrix Associated with matrix inverses is the computation of the determinant of a matrix.Al- though the determinant is a scalar,its definition is a little more complicated than those discussed in the previous paragraphs.Let A be an m x m(square)matrix.The (i,j)- minor of A,denoted Mii,is the determinant of the (m-1)x (m-1)matrix formed by deleting the ith row and the jth column of A.The (i,j)-cofactor of A,denoted Cij,is (-1)Mj.The determinant of a 1x 1 matrix [a],denoted det ([a]),is det ([a])=a. Finally,we define the determinant of an m x m matrix A as m det(A)=∑amC: j=1 In other words,the determinant of a (square)matrix is the sum of the products of the elements in the first row of the matrix and the cofactors of the first row.As is true of inverses,determinants usually are obtained using a computer. Basic Matrix Operations Let c be a real or complex number (often called a scalar).The scalar multiple of scalar c and matrix A,denoted cA,is obtained by multiplying every elements of A by c.If c =-1.the scalar multiple is called the negative of A. Assuming that they have the same number ofrows and columns,the sum oftwo matrices A and B,denoted A+B,is the matrix obtained by adding the corresponding elements2 Chapter 1 A Brief Review of Matrices and Vectors denoted tr(A), is the sum of the elements in the main diagonal of A. Two matrices A and B are equal if and only if they have the same number of rows and columns, and aij = bij for all i and j. The transpose of an m £ n matrix A, denote AT , is an n £ m matrix obtained by interchanging the rows and columns of A. That is, the ®rst row of A becomes the ®rst column of AT , the second row of A becomes the second column of AT , and so on. A square matrix for which AT = A is said to be symmetric. Any matrix X for which XA = I and AX = I is called the inverse of A. Usually, the inverse of A is denoted A¡1 . Although numerous procedures exist for computing the inverse of a matrix, the procedure usually is to use a computer program for this purpose, so we will not dwell on this topic here. The interested reader can consult any book an matrix theory for extensive theoretical and practical discussions dealing with matrix inverses. A matrix that possesses an inverse in the sense just de®ned is called a nonsingular matrix. Associated with matrix inverses is the computation of the determinant of a matrix. Al￾though the determinant is a scalar, its de®nition is a little more complicated than those discussed in the previous paragraphs. Let A be an m £ m (square) matrix. The (i; j)- minor of A, denoted Mij , is the determinant of the (m¡1)£(m¡1) matrix formed by deleting the ith row and the jth column of A. The (i; j)-cofactor of A, denoted Cij, is (¡1) i+jMij. The determinant of a 1£1 matrix [®], denoted det([®]), is det([®]) = ®: Finally, we de®ne the determinant of an m £ m matrix A as det(A) = Xm j=1 a1jC1j : In other words, the determinant of a (square) matrix is the sum of the products of the elements in the ®rst row of the matrix and the cofactors of the ®rst row. As is true of inverses, determinants usually are obtained using a computer. Basic Matrix Operations Let c be a real or complex number (often called a scalar). The scalar multiple of scalar c and matrix A, denoted cA, is obtained by multiplying every elements of A by c. If c = ¡1, the scalar multiple is called the negative of A. Assuming that they have the same number of rows and columns, the sum of two matrices A and B, denoted A + B, is the matrix obtained by adding the corresponding elements