Example Problems of Factor Theorem in Determinants :
Here we are going to see how to solve determinants using factor theorem.
If each element of a matrix A is a polynomial in x and if |A| vanishes for x = a, then (x - a) is a factor of | A |.
(i) This theorem is very much useful when we have to obtain the value of the determinant in ‘factors’ form.
(ii) If we substitute b for a in the determinant | A |, any two of its rows or columns become identical, then | A | = 0, and hence by factor theorem (a - b) is a factor of | A |.
(iii) If r rows (columns) are identical in a determinant of order n (n ≥ r), when we put x = a, then (x - a)r - 1 is a factor of | A |.
(iv) A square matrix (or its determinant) is said to be in cyclic symmetric form if each row is obtained from the first row by changing the variables cyclically.
(v) If the determinant is in cyclic symmetric form and if m is the difference between the degree of the product of the factors (obtained by substitution) and the degree of the product of the leading diagonal elements and if
(1) m is zero, then the required factor is a constant k
(2) m is 1, then the required factor is k(a + b + c) and
(3) m is 2, then the required factor is k(a2 + b2 + c2) + l (ab + bc + ca).
Solve the following problems by using Factor Theorem :
(1) Show that
(2) Show that
(4) Show that
(6) Show that
After having gone through the stuff given above, we hope that the students would have understood, "Example Problems of Factor Theorem in Determinants".
Apart from the stuff given in "Example Problems of Factor Theorem in Determinants", if you need any other stuff in math, please use our google custom search here.
You can also visit our following web pages on different stuff in math.
APTITUDE TESTS ONLINE
ACT MATH ONLINE TEST
TRANSFORMATIONS OF FUNCTIONS
ORDER OF OPERATIONS
Converting customary units worksheet
Customary units worksheet
Integers and absolute value worksheets
Nature of the roots of a quadratic equation worksheets
Trigonometry heights and distances
MATH FOR KIDS
Word problems on linear equations
Trigonometry word problems
Word problems on mixed fractrions
Converting repeating decimals in to fractions