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Determinants are like magic numbers that describe the behaviour of matrices, which are arrays of numbers. Think of them as special codes that reveal secrets about these arrays. One cool trick is their multiplicative property: multiplying two matrices and finding their determinant is like multiplying their secret codes together. Furthermore, switching rows or columns changes the code’s sign. Scalar multiplication is like stretching or shrinking the code, affecting its size. Read on to learn more about the 10 Properties of Determinants along with each of their Formulas and Examples.
Also Read: Order of Operations and PEMDAS Rule
Reflection Property
Table of Contents
The reflection property states that the determinant of a square matrix changes sign when the rows or columns of the matrix are interchanged.
Formula:
Let A be a square matrix, then:
det(A) = -det(A^T)
Example:
Consider the matrix:
A = [1 2]
[3 4]
The determinant of A is:
det(A) = 14 – 23 = -2
Now, if we interchange the rows of A, we get:
A^T = [3 4]
[1 2]
The determinant of A^T is:
det(A^T) = 32 – 41 = 2
Therefore, we can see that det(A^T) = -det(A), which demonstrates the Reflection Property.
Also Read: Conic Sections
All-Zero Property
The All-zero property states that the determinant of a matrix with at least one row or column of all zeros is zero.
Formula:
If a square matrix A has a row or column of all zeros, then det(A) = 0.
Example:
Consider the matrix:
A = [1 2 0]
[3 4 0]
[5 6 0]
The determinant of A is:
det(A) = 0
Thus, this is because the third column of A is all zeros, and the All-zero property applies.
Also Read: Multiplication and Division Word Problems
Proportionality
The Proportionality property states that the determinant of a matrix is proportional to the product of its entries.
Formula:
Let A be a square matrix, then:
det(kA) = k^n * det(A)
where k is a scalar and n is the size of the matrix.
Example:
Consider the matrix:
A = [1 2]
[3 4]
The determinant of A is:
det(A) = 14 – 23 = -2
Now, let’s multiply A by a scalar k = 3:
kA = [3 6]
[9 12]
The determinant of kA is:
det(kA) = 3^2 * det(A) = 9 * (-2) = -18
Hence, this demonstrates the Proportionality property, which is one of the Properties of Determinants.
Switching Property
The Switching property states that the determinant of a matrix is unchanged when any two rows or columns of the matrix are interchanged.
Formula:
Let A be a square matrix, and let A’ be the matrix obtained by interchanging any two rows or columns of A. Then:
det(A) = det(A’)
Example:
Consider the matrix:
A = [1 2 3]
[4 5 6]
[7 8 9]
The determinant of A is:
det(A) = 1*(59 – 68) – 2*(49 – 67) + 3*(48 – 57) = 0
Now, let us interchange the first and second rows of A to get A’:
A’ = [4 5 6]
[1 2 3]
[7 8 9]
The determinant of A’ is:
det(A’) = 4*(29 – 38) – 5*(19 – 37) + 6*(18 – 27) = 0
Therefore, we can see that det(A) = det(A’), hence demonstrating the Switching property.
Also Read: Word Problems on Arithmetic Operations
Factor Property
The Factor property states that the determinant of a matrix is the product of the determinants of its submatrices.
Formula:
Let A be a square matrix, and let A_ij be the submatrix obtained by deleting the i-th row and j-th column of A.
Then:
det(A) = sum_i=1^n (-1)^(i+j) * a_ij * det(A_ij)
Example:
Consider the matrix:
A = [1 2 3]
[4 5 6]
[7 8 9]
The determinant of A can be calculated as:
det(A) = 1*(59 – 68) – 2*(49 – 67) + 3*(48 – 57) = 0
This demonstrates the Factor property, as the determinant of A is the sum of the products of the elements in the first row and the determinants of the corresponding 2×2 submatrices.
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Scalar Multiple Properties
The Scalar Multiple properties state that the determinant of a matrix is linear to its columns or rows.
Formula:
Let A be a square matrix, and let k be a scalar.
Then:
det(kA) = k * det(A)
det(A + kB) = det(A) + k * det(B)
Example:
Consider the matrix:
A = [1 2 3]
[4 5 6]
[7 8 9]
Let k = 2, and let B be the matrix obtained by multiplying the first column of A by 2:
B = [2 2 3]
[8 5 6]
[14 8 9]
The determinant of A is:
det(A) = 1*(59 – 68) – 2*(49 – 67) + 3*(48 – 57) = 0
The determinant of B is:
det(B) = 2*(59 – 68) – 2*(49 – 67) + 3*(48 – 57) = 0
Thus, we can see that det(B) = 2 * det(A), demonstrating the Scalar Multiple property.
Also Read: Unitary Method
Sum Property
The Sum property states that the determinant of a matrix is linear to its columns or rows.
Formula:
Let A and B be square matrices of the same size. Then:
det(A + B) = det(A) + det(B)
Example:
Consider the matrices:
A = [1 2 3]
[4 5 6]
[7 8 9]
B = [0 1 2]
[3 4 5]
[6 7 8]
The determinant of A is:
det(A) = 1*(59 – 68) – 2*(49 – 67) + 3*(48 – 57) = 0
The determinant of B is:
det(B) = 0*(48 – 57) – 1*(38 – 56) + 2*(37 – 46) = -3
The determinant of A + B is:
det(A + B) = 1*(913 – 1012) – 2*(713 – 1010) + 3*(712 – 910) = -3
Therefore, we can see that det(A + B) = det(A) + det(B), demonstrating the Sum property.
Also Read: Profit and Loss Formula Questions
Triangle Property
The Triangle property states that the determinant of a matrix is the product of the lengths of the sides of the triangle formed by the rows or columns of the matrix.
Formula:
Let A be a square matrix, and let a, b, and c be the lengths of the sides of the triangle formed by the rows or columns of A.
Then:
det(A) = a * b * c
Example:
Consider the matrix:
A = [3 4]
[5 12]
The rows of A form a triangle with sides of lengths 3, 4, and 13 (the length of the hypotenuse).
The determinant of A is:
det(A) = 3 * 4 * 13 = 156
Hence, this demonstrates the Triangle property, as the determinant of A is the product of the lengths of the sides of the triangle formed by the rows of A.
Also Read: BODMAS Questions
Determinant of Cofactor Matrix
The determinant of the Cofactor Matrix of a square matrix A is equal to the determinant of A raised to the power of (n-1), where n is the size of the matrix.
Formula:
Let A be a square matrix of size n, and let C be the cofactor matrix of A.
Then:
det(C) = det(A)^(n-1)
Example:
Consider the matrix:
A = [1 2 3]
[4 5 6]
[7 8 9]
The cofactor matrix C of A is:
C = [-3 6 -3]
[6 -12 6]
[-3 6 -3]
The determinant of A is:
det(A) = 1*(59 – 68) – 2*(49 – 67) + 3*(48 – 57) = 0
The determinant of C is:
det(C) = (-3)^2 + (6)^2 + (-3)^2 = 81
We can see that det(C) = det(A)^(n-1), where n = 3, demonstrating the property of the determinant of the Cofactor matrix.
Also Read: Algebra Questions
Property of Invariance
The property of Invariance states that the determinant of a matrix is invariant under elementary row or column operations.
Formula:
Let A be a square matrix, and let A’ be the matrix obtained from A by performing an elementary row or column operation. Then:
det(A) = det(A’)
Example:
Consider the matrix:
A = [1 2 3]
[4 5 6]
[7 8 9]
The determinant of A is:
det(A) = 1*(59 – 68) – 2*(49 – 67) + 3*(48 – 57) = 0
Now, let us perform an elementary row operation by adding 2 times the first row to the second row:
A’ = [1 2 3]
[12 17 21]
[7 8 9]
The determinant of A’ is:
det(A’) = 1*(179 – 218) – 2*(129 – 217) + 3*(128 – 177) = 0
Therefore, we can see that det(A) = det(A’), demonstrating the property of Invariance.
How are Matrix and Determinants Different?
Students usually confuse between matrix and determinants but these few pointers will help you to differentiate between the two. Matrix is an array of numbers or symbols arranged in rows and columns. In a matrix, the number of rows can be different from the number of columns. Moreover they are represented by square brackets.
On the other hand, a Determinant is a single number that is obtained by calculating from a square matrix where the rows and columns are equal. It is a special value that you extract from the table and that tells you something unique about the data or a specific property of the matrix. Moreover they are represented by vertical bars | | or det(A).
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