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Determinants

Lecture on Linear Algebra
Content covered: 
  • This lecture makes you walk through the topics covered in this Linear Algebra chapter.
  • What is Linear Algebra?
    Linear algebra is branch of maths where we study about vectors and its families also known as vector space.
  • Matrix is representation of vector.
  • Topics cover in the following course are:
  1. Determinants
  2. Matrix
  3. Inverse
  4. Rank
  5. Basis and dimension
  6. Eigen Values
  7. Eigen Vectors
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Lecture on Determinants
Content covered: 
  • This lecture explains the concept of Determinants.
  • Determinant is represented by Δ.
  • Determinant will always be of square size.
  • Order of determinant= no. of rows = no. of columns
  •  Δ of order 2  \( \begin{vmatrix}a&b\\c&d\end{vmatrix} \) =ad-bc
  •  Δ of order 3 \( \begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix} \) = \( a \begin{vmatrix}e&f\\h&i\end{vmatrix} -b \begin{vmatrix}d&f\\g&i\end{vmatrix} +c \begin{vmatrix}d&e\\g&h\end{vmatrix}\) 
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Lecture on Properties of Determinants
Content covered: 
  • Important properties of determinant are:
  1. The value of Δ does not change if rows and columns are interchanged.
    |AT|=|A|
    \( \begin{vmatrix}2&3\\4&5\end{vmatrix} \) = \( \begin{vmatrix}2&4\\3 &5\end{vmatrix} \)
  2. If any row or column of a matrix is 0 then |A|=0
    Δ of
    \( \begin{vmatrix}0&0&0&0\\7&5&9&6\\5&3&2&1\\7&6&2&5\end{vmatrix} \) =0
  3. If matrix have identical rows or column . The value of Δ =0. 
    Δ of
    \( \begin{vmatrix}6&5&3\\2&9&7\\6&5&3\end{vmatrix} \) =0
  4. The value of a Δ is unchanged if row or column are added m times.
    A= \( \begin{vmatrix}7&6\\2 &1\end{vmatrix} =-5 \) 
    After R1<-R1+2R2
    \( \begin{vmatrix}11&8\\2 &7\end{vmatrix} =-5 \)
  5. Multiplication with scalar
    A=\( \begin{vmatrix}a1&a2\\ a3 &a4\end{vmatrix} \)  
    A'= \( \begin{vmatrix}k a1&k a2\\ a3 &a4\end{vmatrix} =k|A|\)
  6. Product with cofactor
    \( \begin{vmatrix}a1&a2\\a3 &a4\end{vmatrix} =k^n|A| \) where n is order of Δ
    a1*cf(a1)+a2*cf(a2)=|A|
    a1*cf(a3)+a2*cf(a4)=0
  7. Determinant of multiplication of two matrices =Determinant of first matrix X Determinant of second matrix
      |AB|=|A| X |B|
  8. |Adj A|=|A|n-1
    Adj A=[cofactor A]T
    A .adjA=|A|.I
    |Adj A|=|A|.I/A=|A|n-1 
  9. If any two rows (or two columns )of a determinant are interchanged, the value of the determinant is multiplied by -1.
  10. If all element of a row (or a column ) of a determinant are multiplied by the same number k, the value of the new determinant is k times of the given determinant.
  11. If A be an n-rowed square matrix and 'k' be any scalar then                  |kA|=kn|A|
  12. In a determinant the sum of the product of the element of any row(column) with the cofactors of the corresponding elements of any other row(column) is zero.                                                                                                                                                 Δ= ai1Ai1+ai2Ai2+--------------------------ainAin                                                                          but if ai1Aj1+ai2Aj2+--------------------------ainAjn​=0
  13. In a determinant ,each element in any row (or column ) Consists of the sum of two terms,then the determinant can be expressed as sum of two determinant of the same order.                                                                                               \ \begin{vmatrix}a1+\alpha 1&b1&c1\\a2+\alpha 2&b2&c2\\a3+\alpha 3&b3&c3\end{vmatrix} \\\ \begin{vmatrix}a1&b1&c1\\a2&b2&c2\\a3&b3&c3\end{vmatrix} \\ +\ \begin{vmatrix}\alpha 1&b1&c1\\\alpha 2&b2&c2\\\alpha 3&b3&c3\end{vmatrix} \\
  14. The value of upper triangular or lower triangular matrices or diagonal matrix always be multiplication of diagonal element.
  15. The value of the determinant of skew symmetric matrix of odd order is always zero.
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GATE 1997 question on Determinants

The determinant of the following matrix is :

\(\begin{bmatrix} 6 & -8 & 1 & 1\\ 0 & 2 & 4 & 6\\ 0 & 0 & 4 & 8\\ 0 & 0 & 0 & -1\end{bmatrix}\)

(A) 11

(B) -48

(C) 0

(D) -24

Things you need to know

If A is an upper triangular matrix, then det(A) =  Product of diagonal elements

Therefore,

Determinant = 6*2*4*(-1) = -48

Ans (B)

 

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Cramer's rule

System of non homogeneous Linear equations(Cramer's rule)

 

ax+by+cz=0----homogeneous

ax+by+cz+d=0--------------------non homogeneous

 

Let 'n ' linear simultaneous equation in 'n' unknown x1,x2,x3............xn be

a11x1+a12x2+......................................a1nxn=b1

a21x1+a22x2+......................................a2nxn=b2

.

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.

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an1x1+an2x2+......................................annxn=bn

 

let Δ= \begin{vmatrix} a11&a12 &......a1n & \\ a21 &a22 & ......a2n& \\ .& .& &. \\ an1&an2 &ann & \end{vmatrix}    ≠0

 

suppose A11,A12 ,A13...................etc denotes the cofactors of a11, a12,a13.............. multiply given equation respectively by A11, A21, ................An1and we get 

 

A11(a11x1+a12x2+......................................a1nxn=b1)

A21(a21x1+a22x2+......................................a2nxn=b2)

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.

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An1(an1x1+an2x2+......................................annxn=bn)

 

that is,

         x1(a11A11+ a21A21+......................an1An1​) + x2(0)+x3(0)                            =b1A11+b2A21+..............bnAn1

⇒x1Δ=Δ1

x1=Δ1/Δ

where  Δ1 isobtained by replacing elements in the first column of  Δ bythe elements b1,b2...................bn

Similarly,

we get x2=Δ2/Δ

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.

.

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.xn=Δ2/Δ

 

Example:

solve by Cramer's rule

x+2y+3z=6

2x+4y+z=7

3x+2y+9z=14

 

 

solution

we have to first find the values of Δ i.e cofficent of x ,y,z

Δ=\begin{vmatrix} 6&2 &3& \\ 7 &4 & 1& \\ 14& 2& 9&\\ & \end{vmatrix}\begin{vmatrix} 1&2 &3& \\ 2 &4 & 1& \\ 3& 2& 9&\\ & \end{vmatrix}

 

Δ=-20

 

Δ1\begin{vmatrix} 6&2 &3& \\ 7 &4 & 1& \\ 14& 2& 9&\\ & \end{vmatrix}

=-20

x1

= -20/-20

=1

 

 

Δ2=\begin{vmatrix} 1&6 &3& \\ 2 &7 & 1& \\ 3& 14& 9&\\ & \end{vmatrix}

=-20

y=Δ2

=-20/-20

=1

similarly

Δ3=\begin{vmatrix} 1&2 &6& \\ 2 &4 & 7& \\ 3& 2& 14&\\ & \end{vmatrix}

=-20

z=Δ3/Δ

=1

 

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  • This quiz contains 5 questions on the topic Determinant
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Difficulty Level:  intermediate