(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.


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1 Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product of lower triangular matrices is lower triangular, and the product of upper triangular matrices is upper triangular. (c) A triangular matrix is invertible if and only if its diagonal entries are all non zero. (d) The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular. Symmetric Matrices: A square matrix A is symmetric if A = A T. Example:. Zero matrix and identity matrix are symmetric (any diagonal matrix is symmetric) is a symmetric matrix 5 0 Theorem.7.2,.7.3,.7.: If A and B are symmetric matrices with the same size, and if k is any scalar, then: (a) A T is symmetric (b) A + B and A B are symmetric (c) ka is symmetric (d) The product of two symmetric matrices is symmetric if and only if the matrices commute. (e) If A is an invertible symmetric matrix, then A is symmetric. Chapter 2, Determinants Section 2., Determinants by Cofactor Expansion Importance: We will use determinants for deciding whether a matrix is invertible or not. Also they can be used to write a formula to find inverse of a matrix. Another usage is finding solutions of a linear system of equations. So far we were using a method (forming augmented matrix and row reduction) where we found solutions at the same time, that is to find one of the unknowns we need to solve for each variable. Different from that method by using determinants we can find any one of
2 the unknowns without finding all other variables. [ ] [ ] a b d b Recall: For A = we have A c d = and we said A is invertible ad bc c a if ad bc 0. In fact we call det (A) = ad bc; it can be denoted as A as well. Example: [ ] 3 5. then det = 3 ( 2)5 = = 22 2 [ ] then det = = Now we will look at higher size square matrices and define determinant in general. To define the determinant for n n where n is any nonzero integer, we will use the cofactor expansion. Definition: If A is a square matrix, then the minor of entry a ij is denoted by M ij and is defined to be the determinant of the submatrix that remains when the i th row and j th column of A are deleted. The number C ij = ( ) i+j M ij is called the cofactor of entry a ij. Example: Let A = M = M 2 = M 3 = M 2 = M 22 = M 23 = = 0 ( 2) = 52 C = ( ) + M = = 6 ( 7) = 3 C 2 = ( ) +2 M 2 = = 8 5 = 3 C 3 = ( ) +3 M 3 = = 2 2 = 22 C 2 = ( ) 2+ M 2 = = = 8 C 22 = ( ) 2+2 M 22 = 8 6 = 2 = 3 C 23 = ( ) 2+3 M 23 = 3 2
3 M 3 = M 32 = M 33 = 5 7 = 7 20 = 27 C = 2 = 2 C 3 5 = 0 3 = 3 C 3 = ( ) +3 M 3 = = ( ) 3+2 M 32 = 2 33 = ( ) 3+3 M 3 = 3 Theorem 2..: The determinant of an n n A can be computed by multiplying the entries in any row (or column) by their cofactors and adding the resulting products, that is for each i n and j n det (A) = a j C j + a 2j C 2j a nj C nj (Cofactor expansion along the j th column) det (A) = a i C i + a i2 C i a nj C nj (Cofactor expansion along the i th row) Remark: To find determinant one can choose any row or column to expand. This property is usefull when you have a row or column that has many zeros. Example: A = Calculate det (A). Let s expand along the st row det = ( 2)C + C 2 + C 3 = Let s expand along the st column det = 2C + 3C 2 + C 3 = Remark: For the sign computation that comes from C ij, there is a quick way. To decide how each entry s minor effects determinant in terms of sign, start with + in (,) entry and continue alternating the sign Example: A = Expand along the 2 nd column= Theorem 2..2: If A is an n n triangular matrix (upper triangular, lower triangular, or diagonal), then det (A) is the product of the entries on the main diagonal of the matrix, that is det (A) = a a a nn Example: Let A = 0 7. Then det (A) =
4 Theorem 2.2.2: Let A be a square matrix. Then det (A) = det (A T ) Theorem 2.3.6: If A is an invertible matrix, then Definition: A = det (A) adj(a) If A is any n n matrix and C ij is the cofactor of a ij, the matrix C C 2... C n C =.... C n C n2... C nn is called the matrix of cofactors from A. The transpose of this matrix is called the adjoint of A and is denoted by adj(a). Recall C ij = ( ) i+j M ij Example: Recall the example A = We calculated C = 52, C 2 = 3, C 3 = 3, C 2 = 22, C 22 = 8, C 23 = 3, C 3 = 27, C 32 = 2, C 33 = 3 and det (A) =. Then A = C = adj(a) = adj(a) = det (A) adj(a) = Cramer s Rule: Now we will see the way to calculate a specific solution of a linear system that has a unique solution. Remember we were froming augmented matrix and found each unknown. By Cramer s rule we do not need to find each unknown to find one specific that we want to know. Theorem (Cramer s Rule): If Ax = b is a system of n linear equation in n unknowns such that det (A) 0, then the system has a unique solution. This solution is x = det (A ) det (A), x 2 = det (A 2) det (A),..., x n = det (A n) det (A)
5 where A j is the matrix obtained by replacing the entries in the j th column of A by b b 2 the entries in the matrix b =. Example: Solve the linear system of equations b n 7x 3x 2 = 3 3x + x 2 = 5 [ ] 7 3 Solution: A =, det (A) = = 6 (We will see that this means A is 3 invertible and hence system has a unique solution) By Cramer s rule x = = 9, x 3 5 det (A) 8 2 = = 3 det (A) 8 5
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