Question

linear algebra class

due in 30minutes

please help ASAP!

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example shows the statement is not true in all cases or cite an appropriate statement from the text. (a) To find the determinant of a triangular matrix, add the entries on the main diagonal false, the determinant of a triangular matrix is equal to the number of nonzero entries in the main diagonal, which may not be equal to the sum of the entries in the diagonal. False, the determinant of a triangular matrix is equal to the product of the entries in the main diagonal, which may not be equal to the sum of the entries in the mal diagonal False, the determinant of a triangular matrix is equal to the order of the matrix, which may not be equal to the sum of the entries in the main diagonal, True, the determinant of a triangular matrix is equal to the order of the matrix, which is equal to the sum of the entries in the main diagonal True, the determinant of a triangular matrix is equal to the product of the entries in the main diagonal, which is equal to the sum of the entries in the main diagonal (b) To find the determinant of a matrix, expand by cofactors in any row or column False, to find the determinant of a matrix, expand by cofactors in any row but not in any column. True, to find the determinant of a matrix, expand by cofactors in any row or column false, to find the determinant of a matrix, expand by cofactors in only the first row. False, to find the determinant of a matrix, expand by cofactors in only the first column. False, to find the determinant of a matrix, expand by cofactors in any column but not in any row. (c) When expanding by cofactors, you need not evaluate the cofactors of zero entries. True, in a cofactor expansion each cofactor gets multiplied by the corresponding entry. If this entry is zero, the product will be equal to the cofactor, True, in a cofactor expansion each cofactor gets multiplied by the corresponding entry. If this entry is zero, the product will be zero. True, in a cofactor expansion each cofactor gets multiplied by the corresponding entry. If this entry is zero, the product will be one. False, in a cofactor expansion each cofactor gets multiplied by the corresponding entry. If this entry is zero, the product will be equal to the cofactor. False, in a cofactor expansion each cofactor gets multiplied by the corresponding entry. If this entry is zero, the product will be zero A 90 do 5

Answer 1

(a)If A is triangular (i.e., A has zeros above or below the diagonal), then determinant of A
|A| = product of diagonal elements.

So option (ii) is true.

(b) True

Option (ii)

option it is true Statement is false.. car. The determinant of a hangular matrix equal to the product of the entries in the main diagonal, which may not be equal to the entire sum of the enties in the main deago ral. for suppose A is an uppery triangular matrix . led 19, 001 A 421 922 0 = 1 an (922933 - 0) –o to lower = product of main diag. onal elementa Suppose b) A= lau al 9131 an (A1-911 (922938-922932 -912 (921933-922931 +913 (998 93 2*9 22930 (expanding about first row) - 91 922933-911922932 -912921933+ 9 1292291 2273) +913921932 - 9 292 2931 column 1913 subpose we expand about about third third - 9,8 (912932 - 92 2931) - 923 (911 922-931 912) + Gold 1923 a33 BAR - 91 3922931923 91932 + 923931927 9329 - 913 912 932 - 913 922981 - 929 he can see easily that value of both 20.922 are equal NAVNEET

THEOREM : (Laplace) The determinant of a square matrix A = (al is equal to the sum of the products obtained by multiplying the elements of any row (column) by their respective cofactors: A = 0;141 +,24,2+...+ain | 11 = 2 141; +a;Az; + ... + AnyAn; = ajA;

(c) True

Option (ii) is true.

In a cofactor expansion each cofactor gets multiplied by corresponding entry. If the entry is zero then the product will be zero.