Determinants Learning Objectives: Introduction Properties of Determinants Minors and Cofactors Finding the Area of a Triangle Using Determinants Adjoint and Inverse of a Square Matrix Matrix Method of solving System Linear Equations Examples Introduction If two linear equations, alX bl = 0 and ux b2 = 0, have the same solution, then bl/al or, al/ bl — a2/b2, that is, alb2 ubi 0. Here, alb2 — ubi is known as a deteminant and is denoted by the symbol detA A al bl or by (al b). b2/a2 IAI al, u, bl, b2 are called the elements ofthe detemzinant. There are 2 horizontal rows and 2 vertical columns and so, this is an example of a deteminant of the second order. This second order deteminant has 2! = 2 terms In its expansion, of which one IS positive and the other is negative. The diagonal tem or the leading deteminant is (al b2) whose sign is positive. For three linear equations, alX biy + Cl O, ux b2Y + c2 0 and a-sx b3Y + c3 0, having the same solution we have by cross multiplication, x/ (b2C3 b3C2) y/ (C2a3 — 1 / (a2b3 a3b2) or, x (b2C3 b3C2)/ (a2b3 (C2a3 — (a2b3 — a3b2) and a2b3 Now, the values of x and y must satisfy the first equation. Hence, al (b2C3 b3C2) + bl (C2a3 CI (a2b3 a3b2) — 0. a31)2 O. the expression al b2C3 — alb3C2 a3bIC2 — a2bIC3 a2b3CI — a3b2CI is denoted by the symbol det B IBI al bl a2 b2 c2 or by (al b2 c), having three rows and three columns. This is a deteminant of the third order and has [3 = 6 terms of which three terms are positive and three are negative. Properties of Determinants 1. If the rows and columns In a determinant are Interchanged, then the determinant as a whole remains unchanged. For example,
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al and IBI — detA al al (b2C3 — b3C2) + bl (C2a3 — CI (a2b3 a3b2) Similarly, detB — al al (b2C3 al (b2C3 a2 b3C2) + a2 (b3CI blC3) + a3 (blC2 b2CI) b3C2) + a2b3CI — a2bIC3 a3bIC2 — a3b2CI al (b2C3 — b3C2) + bl (C2a3 — C3&) CI (a2b3 a3b2) — det A 2. The interchange of two rows (or two columns) of a determinant changes the sign of the determinant without changing its numerical value. For example, Let det A a2 b2 c2 and detB— al Now, A And, B det B al al (b2C3 — b3C2) + bl (C2a3 — al a2 I (a2b3 (blC3 b3CI) + b2 (Cla3 — C3aI c2 (alb3 a3b2) a3b1) a2bIC3 — a2 b3CI + a3b2CI — alb2C3 alb3C2 —a3bIc2 al (b2C3 al al bl b3C2) — bl(a3C2 — a2C3) Cl(a3b2 a2 b) det A detA = — detB 3. The value of a determinant is zero when it has two identlcal rows or columns. For example, If we follow property 2, then, Interchanging the Identical two rows or columns of it gives detA = — detA Or 2 detA = 0
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Or detA = 0 4. If a determinant has as its elements, integral functions of x and two of its rows or columns become Identical when x a, then (x — a) IS a factor of the determinant. 5. If every element of a row or column is multiplied by a certain factor, then the determinant is also multiplied by that same factor. For example, al a2 a3 Let A- bl b2 b 3 and A' be the determinant obtained by multiplying all the elements of the first column with x. xal bl Cl xa2 b2 (b2C3 b3C2) + bl (xa2C3 — xa3c2 Cl (xa2b3 xa3b2) xal xa3 b3 c3 x[al b2C3 — alb3C2 a2bIC3 xa1b2c3 — xal b3C2 xa2b1c3 — xa3b1(J2 xa2b3c1 — xa3b2c1 a2b3CI — a3b2CI] xA. a3b1C2 6. If every element of a row or column of a determinant be expressed as the sum or difference of two numbers, then the determinant can be expressed as the sum or difference of two determinants. For example, al bl Let A a2 b2 c2 and A' al + Y bl+y + Y a2 • A'= (al + b3C2) + (bl + a3C2 alb2C3 — alb3C2 yb2C3 — yb3C2 a2bIC3 — a3bIC2 ya2C3 — ya3C2 a2b3CI — a3b2CI + ya2b3 ya3b2 (al b2C3 — alb3C2 a2bIC3 — a3bIC2 a2b3CI — a3b2CI) + (yb2C3 yb3C2 ya2C3 — ya3C2 ya2b3 ya3b2) al bl 7. A determinant remains unchanged by adding or subtracting n times the elements of any row or column to or from the corresponding elements of any other row or column, where n is any given number. For example, al bl A nbl ( nb2 nb3 al + Tibi a2 + nb2 a3 -k nb3 nbl bl Cl nb2 b2 c2 nb3 b3 c3 = O, because the first and second column have similar elements).
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8. The sum of the products of the elements of any row or column of a determinant and the cofactors of the corresponding elements of any other row or column of the determinant IS zero. For example, al a2 a3 Let A bl b2 b3 . Now, let us take the sum of the products of elements of the first row with the cofactors of the corresponding elements of the second row. )2+1 ) 2+2 + —1)2+3 1 al a2(J3 — a3c2 a2 al(J3 — a3cI a3 al(J2 — a2c1 a2a3cI — al a3c2 a2a3cI Minors and Cofactors ala2c3 ala3c2 ala2(J3 In a 3rd order determinant, the minor of an element is obtained by omitting from it the row and the column containing the element in the determinant. al bl Thus, in det A a2 b2 c2 the minors of al, bl and Cl are and respectively. Therefore, we can conclude that the minor of any element in a third order determinant IS a second order determinant. The cofactor of any element in a determinant is its coefficient in the expansion of the determinant and is equal to the corresponding minor with a proper sign. The sign of the cofactor of an ( I)I+J. It is generally denoted by the corresponding element in the i-th row and the j th column IS capital letter. So, in det A, the cofactor of al is ) 1+1 )1+2 for i 1; the cofactor of bl is Bl — ( j -1 -1 1+3 2 b2 for 1 —1, j = 2; and the cofactor of Cl IS Cl ( -1 and so on. Note: A determinant is the sum of the products of the elements of any row or column and the corresponding cofactors. Finding the Area of a Triangle Using Determinants If the coordinates of a triangle are provided than Its area can be derived by using determinants. Thus, if the coordinates of a triangle are (a, b), (n, o) and (x, y), then its area can be found by 1 — [a(o — y) — n(b — y) + x(b — o)] 2
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a 1 2 X b 0 y 1 1 1 Adjoint and Inverse of a Square Matrix If A [atJ]n n be a square matrix, then the transpose of the matrix [Ali] nxnwhose elements are the cofactors of the corresponding elements in IAI is called the adjoint or adjugate matrix of A and is denoted by adj. A. it is equal to [AJl]n n. Thus, all an a13 if A = an an a23 , then adj. A (131 (132 a33 I Al. A 21 A 31 A 12 A 22 A 32 23 where All is the co-factor of at) in A 33 If A and B are square matrices of order n such that AB =BA I, where I is the unit matrix of order n, then B is the inverse of A and is denoted by A l. The inverse of A exists only when A is a non-singular square matrix. Note: singular matrices have adjoints but no Inverse or reciprocals. Matrix Method of solving System Linear Equations The system of simultaneous linear equations alX1 a2X2 a3X3 blX1 b2X2 b3X3 CIXI C2X2 C3X3 al 1
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2. Prove that (a — 1) is a factor of the determinant 3 4 If we put the value a 1 in the above determinant, then it becomes 3 3 4 . Since the first and second columns are the same then the determinant vanishes and it proves that (a 1) is its factor. -1 3. Show that cos C cos B 1 X -1 1 y 1 z 3 z y x 1 X y x z 3 z + xy + x2) (z — x) (z2 1 x 3 x 1 (y (y (y (Z 2 -k ZX -k X 2) + ZX + X2 Y2 — xy — x2) x) {x(z y) (z 2 Y2)} (y x)(z x) (z y 2 4 z) = 0 because (x 3 4 223 y z) cos C -1 cosA cosB cosA -1 cos C cos B cosC cosB -1 cosA cosA -1 R2 + RI COS c R3 + RI cosB — sin 2C -1 O cos C cos C— cosB cosC + 1 cos A cos B cosB cosC + cosA cos B— 1 cosB cosC + cosA cosB cosC + cosA — sin B — [sin 2B sin 2C (cos A + cos B + cos C)2] — [sin 2B sin 2C B cos C} 2] [since A+B+C - [sin 2B sin 2C - {cos (B + cos B C}2] - [Gin B sin 4. Eliminate l, m, n from the equations al + cm bn O, cl + bm + an and express the result in the simplest form. {cos (n B + C)+ cos (sin B sin C) 21 0 and b/ + am cn
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b (a (a Or, a3 + 103 + c 1 c b c (a bc ca ab) (a + 2bc a2 + ab + ca — bc} 3abc) 3abc
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