Properties of Determinants

A determinant is a special number that can be found in a square matrix. Determinants carry important information regarding the matrix and play key roles in many areas of mathematics, especially in linear algebra, calculus, and systems of linear equations.

1.0Determinants of a Square Matrix

The determinant is the scalar quantity attached to each square matrix that yields information about the type of matrix, namely whether it is invertible or not. 

2.0Properties of Determinants

Property 1: Determinant of a Diagonal Matrix 

The determinant is the multiplication or the product of the diagonal elements. For example, given a diagonal matrix $A=diag(a_1,a_2,....,a_n),$

$det(A)=a_1{a}_2......a_n$

Property 2: Determinant of the Identity Matrix 

The determinant of any identity matrix I_n of any order is always equals 1: 

$det(I_n)=1$

Property 3: Determinant of a Triangular Matrix 

In a triangular matrix, whether it is upper or lower triangular, the determinant of such matrix always equals the product of its diagonal elements: 

det(A)=Product of diagonal elements of A

Property 4: Swapping of two Rows/Columns 

In a matrix, if two rows/columns are swapped, the sign of determinant changes: 

det(A')=-det(A)

Property 5: Multiplying a Row/Column by a Scalar 

When a matrix is multiplied by a scalar, say k, the determinant is also multiplied by k: 

$det(kA)=k.det(A)$

Property 6: Adding a multiple of One Row/Column to Another Row/Column

If one row (or column) of a matrix is replaced by the sum of itself and a scalar multiple of another row (or column), then the determinant is unchanged: 

det(A')=det(A)

Property 7: Determinant of an Inverse Matrix 

The inverse of a matrix implies that a matrix, say A, can be invertible, such that its product is always equal to 1. 

$det(A^{-1})=\frac{1}{det(A)}$

Property 8: Determinant of a Product of Matrices 

The determinant of a product of matrices is defined for two square matrices, A and B, of equal size: the determinant of their product is equal to the product of individual determinants:

det(AB)=det(A).det(B)

Property 9: Determinant of a Matrix with a Row or Column of Zeroes

A matrix has a zero determinant if elements of two rows(or columns) are equal, the matrix is singular and lacks an inverse; its determinant is zero.

det(A) =0

Property 10: Determinant of a Matrix with a Row or Column of Zeroes 

The determinant of any matrix whose row or column is zero will always be zero. This is because a matrix with a row or column consisting wholly of zeros is singular and does not have a unique solution to the associated system of equations. 

det(A) = 0

To get a better understanding of the topic, let’s explore the properties of determinants with examples that are useful in performing important mathematical operations on determinants. 

3.0Solved Examples 

Problem 1: Find the determinant of the matrix: 

  A = $\left|\begin{array}{ccc}2& 3& 1\\ 4& 5& 6\\ 7& 8& 9\end{array}\right|$

Solution: Let’s expand the determinant along the first row: 

$det(A)=2\times \left|\begin{array}{cc}5& 6\\ 8& 9\end{array}\right|-3\times \left|\begin{array}{cc}4& 6\\ 7& 9\end{array}\right|+1\times \left|\begin{array}{cc}4& 5\\ 7& 8\end{array}\right|$

$det(A)=2(5\times 9-8\times 6)-3(4\times 9-7\times 6)+1(8\times 4-5\times 7)$

$det(A)=2(45-48)-3(36-42)+(32-35)=2(-3)-3(-6)+(-3)$

$det(A)=-6+18-3=18-9=9$

Problem 2: Find the determinant of the matrix: 

B = $\left|\begin{array}{ccc}2& 4& 6\\ 1& 3& 5\\ 0& 1& 2\end{array}\right|$

Solution: Performing row operation to simplify the given matrix 

Subtract the 2 multiplied by the first row from the second to get a new second row: 

$R_2\to R_1-2R_2$,

$\left|\begin{array}{ccc}2& 4& 6\\ -1& -1& -1\\ 0& 1& 2\end{array}\right|$

Now, add the first row to the third row to get a new third row: 

$R_3\to R_3+R_1$

$\left|\begin{array}{ccc}2& 4& 6\\ -1& -1& -1\\ 2& 5& 8\end{array}\right|$

Now, calculate the determinant by expanding along the first row: 

$det(B)=2\left|\begin{array}{cc}-1& -1\\ 5& 8\end{array}\right|-4\left|\begin{array}{cc}-1& -1\\ 2& 8\end{array}\right|+6\left|\begin{array}{cc}-1& -1\\ 2& 5\end{array}\right|$

$det(B)=2(8\times -1-5\times -1)-4(8\times -1-2\times -1)+6(5\times -1-2\times -1)$

$det(B)=2(-3)-4(-5)+6(-3)$

$det(B)=-6+20-18=0$

Problem 3: Find the determinant of the matrix: 

C = $\left|\begin{array}{ccc}3& 1& 2\\ 2& 3& 4\\ 1& 2& 3\end{array}\right|$

Solution: Here, we will perform a column operation to simplify the matrix: 

Subtract 2 multiplied by the first column from the second column: 

$C_2\to C_2-2C_1$

$\left|\begin{array}{ccc}3& -5& 2\\ 2& -3& 4\\ 1& 0& 3\end{array}\right|$

Calculate the simplified matrix along the first row 

$det(C)=3\times \left|\begin{array}{cc}-3& 4\\ 0& 3\end{array}\right|-(-5)\times \left|\begin{array}{cc}2& 4\\ 1& 3\end{array}\right|+2\times \left|\begin{array}{cc}2& -3\\ 1& 0\end{array}\right|$

$det(C)=3(-9-0)+5(6-4)+2(0+3)$

$det(C)=-27+10+6=-27+16=-11$