Properties of Determinants

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 the 2 × 2 Matrix. The Determinants are calculated by

Det \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix}  \]

The larger Matrices have more complex formulas.

Determinants have 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 that is equal to 0. The Determinant is also used in multiple variable calculi (mainly in Jacobina) and in computing the cross product of vectors.

How is a Determinant different from a Matrix?

This is a question which many students keep pondering upon and also end up mixing the concepts in the exam and losing marks. Though both of them have importance in practical terms, the major key differences between the two are:

1. In a Matrix, a set of numbers are enclosed in a bracket whereas in a Determinant numbers are enclosed in two bars

2. The number of rows and columns in a Matrix is always the same. This is not true for the Determinants

3. Determinants help in determining the values of unknown variables using Cramer’s rule whereas Matrices are used for Mathematical operations such as addition, subtraction, etc. 

Properties of Determinants - Explanation, Important Properties, Solved Examples and FAQs

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) = \[\frac{1}{det (X)}\] = 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. These properties make calculations easier and also are helping in solving various kinds of problems. The description of each of the 10 important properties of Determinants is given below.

1. Reflection Property

2. All-zero Property

3. Proportionality

4. Switching property

5. Factor property

6. Scalar multiple properties

7. Sum property

8. Triangle property

9. Determinant of cofactor Matrix

10. Property of Invariance

Each of these properties is discussed in detail below:

1. Reflection Property

The reflection property of Determinants defines that Determinants do not 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 \[\Delta\] becomes 0 while considering the value of x  = α, then (x -α) is considered as a factor of \[\Delta\].

6. Scalar Multiple Property

If all the elements of a row (or columns) of a Determinant are 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 comprises 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

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

In the above Determinants of the cofactor Matrix, Cij denotes the cofactor of the elements aij in \[\Delta\].

10. Property of Invariance

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

It implies that Determinant remains unchanged under an operation of the term \[C_{i} \rightarrow C_{i} + \alpha C_{j} + \beta C_{k} \] where, j and k is not equivalent to i, or a Mathematical operation of the term \[R_{i} \rightarrow R_{i} + \alpha R_{j} + \beta R_{k} \], where, j and k is not equivalent to i.

Examples Problems on Properties of Determinants

Some important examples on properties of Determinants are given below:

1. Using Properties of Determinants, Prove That

\[ \begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{3}\\ a_{3} & b_{3} & c_{3} \end{vmatrix} = (x + y + z)  (xy + yz + zx - x^{2} - y^{2} - z^{2}) \]

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

\[ \Delta =  \begin{vmatrix} x & y & z \\ y & z & x \\ z & x & y \end{vmatrix} = \begin{vmatrix} x + y + z & y & z \\ y + z + x & z & x \\ z + x + y & z & y \end{vmatrix}\] Operating\[ C_{[1]} \rightarrow C_{[1]} + C_{[1]} + C_{[1]}\]

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

(Operating\[ R_{[2]} \rightarrow R_{[2]} - R_{[1]} and R_{[3]} \rightarrow R_{[3]} - R_{[1]} \])

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

\[ = (x + y + z)(xy + yz + zx - x^{2} - y^{2} - z^{2}) \]

2.  Using Properties of Determinants, 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 the 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

Fun Fact: You might find it very interesting to know that Determinants were introduced by the great Mathematician and physicist Gauss in his book Disquisitiones arithmeticae while talking about quadratic equations in 1801. He devised it as a way of determining solutions for the quadratic equations. Thus, it is a very old concept and continues to hold such a high level of importance to this date. This is the beauty of Maths, it never gets old!  

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 the Matrix

3. Bracket around Matrix

4. None of the above

3. The Matrix product XY= O, then

1. X = O and Y = O

2. X = O or Y = O

3. X is a null Matrix

4. None of the above 

4. If Z  is a unit Matrix, then 3Z will be

1. a unit Matrix

2. a triangular Matrix

3. a scalar Matrix

4. None of the above