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102 LearnersLast updated on September 1, 2025
Determinant of a Matrix
The determinant of a square matrix is a scalar value derived from its elements. This value provides insights into various properties of the matrix. A singular matrix has a zero determinant, whereas a non-singular matrix has a non-zero determinant. Geometrically, the absolute value of the determinant represents a scaling factor for areas or volumes when the matrix is used to transform shapes. Determinants are used in solving linear equations and analyzing linear transformations.
What is the Determinant of a Matrix
In mathematics, the determinant is a special value calculated from a square matrix that tells us important properties, such as whether the matrix is invertible. It also helps in solving systems of linear equations and understanding how the matrix transforms space. It plays a key role in linear algebra and geometry.
What are the Properties of the Determinant of a Matrix
The properties of the determinant of a matrix describe how the determinant behaves under various operations like row swaps, scaling, and matrix multiplication. These rules help to solve linear systems and understand the geometric effect of matrices.
1. Determinants are defined exclusively for square matrices
You cannot calculate the determinant of a rectangular matrix (2×3 or 3×2). Determinants are used to study and solve systems of equations, both of which require square matrices.
2. When two rows or columns of a matrix are swapped:
The determinant’s magnitude remains the same, but its direction flips. It is used in geometry and linear transformations, as it helps in understanding how changes in the matrix affect the spatial properties of the transformed object.
3. If two rows or columns are identical, the determinant is zero
If two rows or columns in a matrix are identical, the determinant equals zero. This is because swapping identical rows or columns does not change the matrix. The rows and columns of these matrices are linearly independent. Therefore, it is a singular matrix.
4. If a row or column is all zeros, the determinant is zero
The matrix maps the space into a lower dimension, which indicates singularity. Here, the matrix collapses space into a lower dimension, making it non-invertible. In other words, the matrix does not have full dimensions of the space, leading to loss of information. Therefore, there is no inverse matrix as well.
5. Multiplying all elements of a row or column by a constant scales the determinant by that same constant.
If you multiply a single row (or column) by a number, that number also multiplies the determinant. This demonstrates how scalar multiplication influences the scaling effect of the matrix's transformation.
6. The determinant of the identity matrix equals 1. This holds for identity matrices of any size,
Here the determinant is always 1, meaning it does not change any vector it’s applied to. It is the neutral element in matrix multiplication.
7. If one row of a matrix is a scalar multiple of another row, the determinant of the matrix is zero.
If any row (or column) is just a scaled copy of another, the matrix is not independent. This means the matrix is singular, and you cannot find its inverse.
8. Determinant of a product equals the product of the determinants
For two square matrices of the same size:
det(AB)=det(A). det(B). This helps in simplifying large matrix problems and understanding composite transformations.
9. The determinant of a transpose is equal to the determinant of the original matrix
Transposing a matrix means flipping rows into columns and vice versa. The determinant remains the same:
det(AT) =det(A). This shows symmetry in determinant behavior — useful in proofs and theoretical math.
10. If the determinant is zero, the matrix is not invertible
A matrix with a determinant of zero is called a singular matrix. It cannot be reversed, there’s no inverse.
What is the Minor of an Element of a Matrix
What is the Minor of an Element of a Matrix
The minor of an element in a matrix is the determinant of the smaller matrix obtained by removing the row and column that contain that element. This concept is essential for calculating cofactors and expanding the determinant of larger matrices. Minors are essential in matrix operations like finding inverses and solving systems. If you have a square matrix, and you choose a particular element from it, the minor of that element is found by:
Removing the entire row in which the element is located.
Removing the entire column in which the element is located.
Taking the remaining smaller square matrix, and
Calculating its determinant, that number is called the minor of the original element.
Example (3×3 Matrix):
Let’s take a matrix:
To find the minor of element 5 (which is in the 2nd row, 2nd column):
Remove row 2 and column 2:
Now, find the determinant of that 2×2 matrix:
(1×9)−(3×7)=9−21=−12 is the final answer
What is a Cofactor of an Element of a Matrix
The cofactor of an element in a matrix is a number that is used to help calculate the determinant and the inverse of a matrix. It is closely related to the minor of the element but includes a sign adjustment based on the element's position.
Definition in Simple Words:
The cofactor of an element is:
The minor of that element, multiplied by either +1 or –1, depending on its position in the matrix.
This sign is determined using a checkerboard pattern of plus and minus signs.
Cofactor Formula:
The cofactor of an element aij in a square matrix is calculated as
Cij=(-1)i+j . mij
Where:
i and j are the row and column indices of the element.
Mij is the minor of aij which is the determinant of the matrix obtained by removing the i-th row and j-th column from the original matrix
Determinant of a 2x2 Matrix
The determinant of a 2×2 matrix is a simple but powerful value that helps analyze the properties of the matrix. This calculation determines whether the matrix is invertible and indicates how the matrix scales areas during geometric transformations.
A=
Formula to Find the Determinant:
det(A)=(ad)−(bc)
You multiply the top-left and bottom-right elements (a and d), then subtract the product of the top-right and bottom-left elements (b and c).
Example:
A=
det(A)=(4×1)−(3×2)=4−6=−2
So, the determinant is –2.
Determinant of a 3 x 3 Matrix
The determinant of a 3 × 3 matrix is a value that indicates whether the matrix is invertible and how it transforms space. It is calculated using a specific formula involving the elements of the matrix. This determinant is essential in solving systems of equations, finding inverses, and understanding 3D transformations.
General Form of a 3 × 3 Matrix:
A=
Formula to Find the Determinant:
det(A)=a(ei−fh)−b(di−fg)+c(dh−eg)
This is done by expanding along the first row using minors and cofactors.
Example:
Let’s find the determinant of:
A=
Step-by-step:
det(A)=1(5×9−6×8)−2(4×9−6×7)+3(4×8−5×7)
=1(45−48)−2(36−42)+3(32−35)
=1(−3)−2(−6)+3(−3)
=−3+12−9=0
So, the determinant is 0, which means the matrix is singular and not invertible.
What are the Formulas for the Determinants of a Matrix
The formulas for the determinants of a matrix vary based on the matrix size. For a 2 × 2 matrix, it's a simple difference of products. For 3 × 3 or larger, it involves minors and cofactors. Determinant formulas help in matrix inversion, solving linear equations, and understanding geometric transformations in space.
1. Determinant of a 2 × 2 Matrix
For a matrix:
A=
The formula is: det(A)=(a⋅d)−(b⋅c)
2. Determinant of a 3 × 3 Matrix
For a matrix:
A=
The formula is: det(A)=a(ei−fh)−b(di−fg)+c(dh−eg)
This is called cofactor expansion along the first row.
3. Determinant of a 4 × 4 or Higher Matrix
To find the determinant of a 4 × 4 matrix or larger, you use a method called cofactor expansion
Let's solve this
Now we will use cofactor expansion along the first row:
det(A) = a11c11 - a12 c12 + a13 c13 - a14c14
Now, we will choose any row or column, usually we choose the one with the most zeros to simplify.
Where each c1j is the determinant of the 3 3 minor matrix formed by removing the first row and the j-th
Real-Life Applications of Determinant of a Matrix
Determinants of matrices are used in solving linear equations, eigenvalues, and eigenvectors. In the fields of economics, physics, computer science, this property is used massively.
Cryptography: Encoding and decoding: Determinants play a key role in encryption methods such as the Hill Cipher, which relies on matrices. For the encoded message to be decrypted, the matrix used must be invertible, meaning its determinant must not be zero.
Robotics and Control Systems: Kinematics: Determinants help determine whether a robotic arm’s movement equations are solvable.
System stability: In control theory, the determinant of certain matrices (like Jacobians) is used to study system behavior and stability.
Geography and Navigation: Coordinate transformations: Determinants are used to convert between different geographic coordinate systems, like mapping the line.
Area and volume calculation: Determinants help compute areas of triangles and volumes of 3D shapes from coordinate data.
Data Science and Machine Learning: Feature independence: Determinants are used in linear algebra to check if feature vectors are linearly independent. Multivariate statistics: Covariance matrices use determinants to assess data spread and relationships between variables.
Engineering and Physics: Structural analysis: Determinants help solve equations that model stresses and forces in structures like bridges and buildings.
Electric circuits: Kirchhoff’s laws involve systems of equations where determinants are used to find unknown currents and voltages.
Mechanical systems: In dynamics and motion equations, determinants help solve systems related to force and movement.
Common Mistakes of Determinant of a Matrix
The most common mistakes made by students while practicing the determinants of matrices are wrong use of signs and incorrect application of row and column functions. Even in finding cofactor students also make mistakes. Here are some examples of such mistakes noted below:
Mistake 1
Applying Determinant to Non-Square Matrices
While solving the determinant of a matrix that is not square, like a 2×3 or 3×2 matrix, this is the mistake students often make. The determinant can only be defined for square matrices; the number of rows must be equal to the number of columns, 2×2, 3×3. A matrix should be square (n×n) to have an actual determinant. Non-square matrices make mistakes in the necessary structure for determinant-based calculations
A= 2 3, which is incorrect
As this matrix has 2 rows and 3 columns, it’s not square, so you cannot find its determinant.
B= 2 2
det(B) = (2)(3) - (4)(1) = 6–4 = 2
Now we have a matrix that is square (2 × 2), so calculating the determinant is valid.
Mistake 2
Wrong Minor Matrix Selection
When calculating the determinant using cofactor expansion, each element is associated with a minor, which is the determinant of a smaller matrix formed by deleting the element's row and column. Always double-check that you're selecting the correct row and column before calculating the minor.
Mistake 3
Ignoring Sign Change When Swapping Rows
Swapping two rows, but not changing the sign of the determinant. In determinant rules, each row swap reverses the sign of the determinant. Ignoring this leads to incorrect results. Every time you swap two rows (or columns), multiply the determinant by -1. Keep track of how many swaps you perform.
Mistake 4
Misusing Determinant Properties
Assuming the determinant is additive, det(A + B) = det(A) + det(B), or det(AB) = det(A) + det(B). The determinant is not a linear function over matrix addition. While det(AB) = det(A) × det(B) it is true,
det(AB) = det(A) × det(B)
det(A⁻¹) = 1/det(A) (if A is invertible)
det(A^T) = det(A)
det(cA) = cⁿ × det(A)
for a n×n matrix and scalar c
Mistake 5
Ignoring Determinant = 0 Implication
Not realizing that a matrix with a zero determinant is non-invertible and singular, and trying to find its inverse anyway. A zero determinant means the matrix’s rows or columns are linearly dependent. Therefore, it doesn’t have full rank and cannot be inverted. Before attempting to compute an inverse or use Cramer’s Rule, always check if det(A) ≠ 0. If it equals zero, you must use alternative methods like pseudo-inverses.
Solved Examples on Determinant of a Matrix
Problem 1
2×2 Matrix, Find the determinant of the matrix
A=
det(A)=(5)(2)−(3)(7)=10−21=−11
Explanation
For a 2 × 2 matrix
The determinant is ad−bc.
Here, a=5, b=3, c=7, d=2.
Problem 2
3 × 3 Matrix (Cofactor Expansion)
A=
Now we use cofactor expansion along with the first row:
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
Explanation
This method involves multiplying each element by the determinant of its minor and applying the correct sign.
Problem 3
Zero Determinant
Find the determinant of the matrix
C=
det(C)=(4)(4)−(8)(2)=16−16=0
Explanation
The rows are proportional; the second row is half of the first, so the matrix is singular. A determinant of 0 means it is not invertible.
Problem 4
Determinant Using Row Swap
Original determinant: (1)(4)−(2)(3))=4−6=−2
After the row swap, the determinant becomes 2.
Explanation
Swapping two rows reverses the sign of the determinant. So det(D′)=−(−2)=2.
Problem 5
Triangular Matrix Determinant
Find the determinant of the matrix
E=
det(E)=3⋅5⋅(−2)=−30
Explanation
This is an upper triangular matrix (all values below the diagonal are 0).
For any triangular matrix, the determinant is the product of the diagonal elements:
3⋅5⋅(−2)=−30
FAQs of Determinant of a Matrix
1.What is the determinant of a matrix?
2.Can we find the determinant of a non-square matrix?
3.What does it mean if the determinant is zero?
4.Why are determinants important?
5.How is the determinant of a 3×3 matrix calculated?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
