Properties of Determinants

Determinants and Its Properties

In Linear algebra, a determinant is a unique number that can be ascertained from a square matrix. The determinants of a matrix say K is represented as det (K) or,  |K| or det K. The determinants and its properties are useful as they enable us to obtain the same outcomes with distinct and simpler configurations of elements. The determinant is considered an important function as it satisfies some additional properties of determinants that are derived from the following conditions.

  1. Multiplicativity; det (XY) = det (X) det (y)

  2. Invariance under transpose det (X) = det (Xt).

  3. Invariance under row operations; if  X’ is a matrix formed by summing up the multiple of any row to another row, then det (X) = det (X’).

  4. There is a change of sign under row swap. If X’ is a matrix made by interchanging the positions of two rows, then det (X’) = -det (x)


What is known as Determinants ?

The determinant of a square matrix is a value ascertained by the elements of a matrix. In 2 × 2 matrix. The determinants is calculated by

Det\[\begin{pmatrix}a & b\\ c & d\end{pmatrix}\] = ad - bc


The larger matrices have more complex formulas..

Determinants have various different applications throughout Mathematics. For example, they are used in shoelace formulas for calculating the area which is beneficial as a collinearity condition as three collinear points define a triangle which is equal to 0. The determinant is also used in multiple variable calculus(mainly in Jacobina) and in computing the cross product of vectors.


Basic Properties of Determinants 

Some basic properties of determinants are given below:

  1. If In is the identity matrix of the order m ×m, then det(I) is equal to1

  2. If the matrix XT is the transpose of matrix X, then det (XT) = det (X)

  3. If matrix X-1 is the inverse of matrix X, then det (X-1) = 1/det (x) = det(X)-1 

  4. If two square matrices x and y are of equal  size, then det (XY) = det (X) det (Y)

  5. If matrix X retains size a × a and C is a constant, then det (CX) = Ca det (X)

  6. If A, B, and C are three positive semidefinite matrices of equal size, then the following equation holds along with the corollary det (A+B) ≥ det(A) + det (B) for A,B, C ≥ 0 det (A+B+C) + det C ≥ det (A+B) + det (B+C)

  7. In a triangular matrix, the determinant is equal to the product of the diagonal elements.

  8. The determinant of a matrix is zero if each element of the matrix is equal to zero.

  9. Laplace’s Formula and the Adjugate Matrix.

 

Important Properties of Determinants

There are 10 important properties of determinants that are widely used. The description of each of the 10 important properties of determinants are given below.


1. Reflection Property

The reflection property of determinants defines that determinants do no change if rows are transformed into columns and columns are transformed into rows.


2. All- Zero Property

The determinants will be equivalent to zero if each term of rows and columns are zero.


3. Proportionality (Repetition Property)

If each term of rows or columns is similar to the column of some other row (or column) then the determinant is equivalent to zero.


4. Switching Property

The interchanging of any two rows (or columns) of the determinant changes its signs.


5. Factor Property

If a determinant Δ beomes 0 while considering the value of x  = α, then (x -α) is considered as a factor of Δ.


6. Scalar Multiple Property

If all the elements of a row (or columns) of a determinant is multiplied by a non-zero constant, then the determinant gets multiplied by a similar constant.


7. Sum Property

\[\begin{vmatrix}j_{1}+k_{1} & l_{1} & m_{1} \\ j_{2}+k_{2} & l_{2} & m_{2}\\ j_{3}+k_{3} & l_{3} & m_{3}\end{vmatrix}\] = \[\begin{vmatrix}j_{1} & l_{1} & m_{1} \\ j_{2} & l_{2} & m_{2}\\ j_{3} & l_{3} & m_{3}\end{vmatrix}\] + \[\begin{vmatrix}+k_{1} & l_{1} & m_{1} \\ +k_{2} & l_{2} & m_{2}\\ +k_{3} & l_{3} & m_{3}\end{vmatrix}\]


8. Triangle Property

If each term of a determinant above or below the main diagonal comprise zeroes, then the determinant is equivalent to the product of diagonal terms. That is

\[\begin{vmatrix}x_{1} & x_{2} & x_{3} \\ 0 & y_{2} & y_{3}\\ 0 & 0 & z_{3}\end{vmatrix}\] = \[\begin{vmatrix}x_{1} & 0 & 0\\ x_{2} & y_{2} & 0 \\ x_{3} & y_{3} & z_{3}\end{vmatrix}\] = X\[_{1}\]Y\[_{2}\]Z\[_{3}\]


9. Determinant of Cofactor Matrix

Δ = \[\begin{vmatrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23}\\ x_{31} & x_{32} & x_{33}\end{vmatrix}\] then Δ\[_{1}\] = \[\begin{vmatrix}z_{11} & z_{12} & z_{13} \\ z_{21} & z_{22} & z_{23}\\ z_{31} & z_{32} & z_{33}\end{vmatrix}\] = Δ\[^{2}\]

In the above determinants of the cofactor matrix,Cij denotes the cofactor of the elements aij in Δ.


10. Property of Invariance

\[\begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2}\\ a_{3} & b_{3} & c_{3}\end{vmatrix}\] = \[\begin{vmatrix}a_{1}+\alpha b_{1}+\beta c_{1} & b_{1} & c_{1} \\ a_{2}+\alpha b_{2}+\beta c_{2} & b_{2} & c_{2}\\ a_{3}+\alpha b_{3}+\beta c_{3} & b_{3} & c_{3}\end{vmatrix}\] 

It implies that determinant remains unchanged under an operation of the term Ci ⟶ Ci + αCj + βCkj where, j and k is not equivalent to i, or a Mathematical operation of the term Ri ⟶ Ri + αRj  + βRk, where, j and k is not equivalent to i.


Examples Problems on Properties of Determinants

Some important example on properties of determinants are given below:


1. Using Properties of Determinant, Prove That

\[\begin{vmatrix}x & y & z\\ y & z & x\\ z & x & y\end{vmatrix}\] = (x + y + z)(xy + yz + zx - x² - y² - z²)


Solution:  With the help of the invariance and scalar multiple properties of determinant we can prove the above- given determinant.

Δ = \[\begin{vmatrix}x & y & z\\ y & z & x\\ z & x & y\end{vmatrix}\] = \[\begin{vmatrix}x+y+z & y & z\\ y+z+x & z & x\\ z+x+y & z & y\end{vmatrix}\] [Operating C\[_{1}\] ⟶ C\[_{1}\] + C\[_{2}\] + C\[_{3}\]]

=  (x + y + z) \[\begin{vmatrix}1 & y & 0\\ 1 & z & x\\ 1 & x & y\end{vmatrix}\]

= (x + y + z) \[\begin{vmatrix}1 & y & z\\ 0 & z-y & x-z\\ 1 & x-y & y-z\end{vmatrix}\][Operating (R\[_{2}\] ⟶ R\[_{2}\] - R\[_{1}\] and (R\[_{3}\] ⟶ R\[_{3}\] - R\[_{1}\])]

= ( x + y + z) [(z - y)(y - z)- (x - y) (x - z )

= ( x + y + z) (xy + yz + zx - x² - y² - Z²)


2.  Using Properties of Determinant, Prove That

\[\begin{vmatrix}a & b & c\\ d & e & f\\ g & h & i\end{vmatrix}\] = \[\begin{vmatrix}b & h & e\\ a & g & d\\ c & i & f\end{vmatrix}\]


Solution: Interchanging the rows and columns across the diagonals by making use of reflection property and then using the switching property of determination we can get the desired outcome.

L.H.S = \[\begin{vmatrix}a & b & c\\ d & e & f\\ g & h & i\end{vmatrix}\] = \[\begin{vmatrix}a & d & g\\ b & e & h\\ c & f & i\end{vmatrix}\]

(Interchanging rows and columns across the diagonals)

= (-1)\[\begin{vmatrix}a & g & d\\ b & h & e\\ c & i & f\end{vmatrix}\] = (1)² = \[\begin{vmatrix}b & h & e\\ a & g & d\\ c & i & f\end{vmatrix}\] = \[\begin{vmatrix}b & h & e\\ a & g & d\\ c & i & f\end{vmatrix}\] = R.H.S


Quiz Time

1. According to the Determinant Properties, the Value of Determinant Equals to Zero if Row is

  1. Multiplied by row

  2. Multiplied to column

  3. Divided to row

  4. Divided to column


2. The Determinants of Matrix in Matrices is Represented By

  1. Vertical lines around the matrix.

  2. Horizontal lines around matrix

  3. Bracket around matrix

  4. None of the above

FAQ (Frequently Asked Questions)

1. When Can We Get the Determinant of a Matrix Equivalent to Zero?

The determinants of a matrix will be equivalent to 0 under the following situations:

  1. Each term of a row is zero.

  2. Two rows and column are similar

  3. A row or column is a constant multiple of other row or columns

A matrix is invertible, nonsingular if and only if the value of determinant is not equal to zero, So if the determinant is zero, the matrix is singular and does not have an inverse.

2. What is the Key Difference Between Matrices and Determinants?

Both matrices and determinants are part of Mathematics. Both play an important role in line equations and also used to solve real-life problems in Physics, Mechanics and Optics etc.

The key difference between matrix and determinants are given below:

  1. The matrix is a set of numbers that are enclosed by two brackets whereas the determinants is a set of numbers that are enclosed by two bars.

  2. The number of rows is always equivalent to the number of columns in the matrix whereas in determinant the number of rows is not equal to the number of columns.

  3. The matrix can be used for operating mathematical operations such as addition, subtraction or multiplication whereas determinants are used for calculating the value of variables such as x,y, and z through Cramer's rule.