Can every matrix have a determinant?

No. In the standard school-level treatment, a determinant is defined for a square matrix only.

What is the determinant of a 2 by 2 matrix?

For A = [[a,b],[c,d]], det(A) = ad - bc.

When is a matrix singular?

A square matrix is singular when its determinant is zero.

When does the inverse of a matrix exist?

A square matrix has an inverse exactly when its determinant is non-zero.

Mathematics • Matrices • Determinants

What Are Determinants? And All Important Determinant Formulas

Q: What is a determinant?

A determinant is a single number associated with a square matrix. It is written as det(A) or |A| and is useful for solving equations, testing invertibility and many other applications.

A determinant is a special number associated with a square matrix. It helps us evaluate matrices, solve systems of equations, find areas, test whether an inverse exists and understand important properties of linear transformations.

In simple words: A determinant converts a square matrix into one number that carries important information about the matrix.

1. What Is a Determinant?

For a square matrix , its determinant is a single number denoted by , , or sometimes .

A=\begin{bmatrix}a&b\\c&d\end{bmatrix}, \qquad |A|=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc

Determinants are defined for square matrices. In the Class 12 NCERT treatment, determinants are studied up to order 3, together with their properties, minors, cofactors, applications to area, adjoint, inverse and systems of linear equations.

2. Why Are Determinants Important?

They help calculate whether a square matrix is singular or non-singular.
They tell us whether a matrix has an inverse.
They provide a method for solving systems of linear equations.
They help calculate the area of a triangle using coordinates.
They are used to define minors, cofactors and the adjoint.
They appear in coordinate geometry, linear algebra, engineering, physics and computer science.

Most important connection: means the square matrix is singular and has no ordinary inverse. If , the matrix is non-singular and is invertible.

3. Determinant of Order 1

For a one-element square matrix:

A=[a]\quad\Rightarrow\quad |A|=a

4. Determinant of Order 2

For

A=\begin{bmatrix}a&b\\c&d\end{bmatrix}

the determinant is:

|A|=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc

Memory rule: For a 2 × 2 determinant, multiply the main diagonal and subtract the product of the other diagonal.

5. Determinant of Order 3

For

A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}

expansion along the first row gives:

|A|=a(ei-fh)-b(di-fg)+c(dh-eg)

Equivalently:

|A|=a(ei-fh)-b(d i-f g)+c(d h-e g)

The determinant can be expanded along any row or any column. When choosing an expansion, a row or column containing zeros often makes the calculation shorter.

6. Minors

The minor of an element is obtained by deleting the row and column containing that element and taking the determinant of what remains.

For the element , its minor is:

M_{ij}=\text{determinant obtained after deleting the }i\text{-th row and }j\text{-th column}

For example, in a 3 × 3 determinant, the minor of is the determinant left after deleting row 1 and column 2.

7. Cofactors

The cofactor of is:

C_{ij}=(-1)^{i+j}M_{ij}

The sign pattern for cofactors is:

\begin{bmatrix}+&-&+\\-&+&-\\+&-&+\end{bmatrix}

Thus, cofactors use alternating signs beginning with positive at the top-left position.

8. Expansion of a Determinant Using Cofactors

For a 3 × 3 determinant, expansion along the first row is:

|A|=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}

More generally, for an n × n determinant, expansion along row i is:

|A|=\sum_{j=1}^{n}a_{ij}C_{ij}

Expansion along column j is:

|A|=\sum_{i=1}^{n}a_{ij}C_{ij}

9. Important Properties of Determinants — Complete Revision List

Property 1: Interchange of two rows or columns

Interchanging any two rows or any two columns changes the sign of the determinant.

R_i\leftrightarrow R_j\quad\Rightarrow\quad |A|\to-|A|

Property 2: Two equal rows or columns

If two rows or two columns are identical, the determinant is zero.

R_i=R_j\quad(i\neq j)\quad\Rightarrow\quad |A|=0

Property 3: Proportional rows or columns

If two rows or two columns are proportional, the determinant is zero.

R_i=kR_j\quad(i\neq j)\quad\Rightarrow\quad |A|=0

Property 4: Zero row or column

If every element of a row or column is zero, the determinant is zero.

R_i=0\quad\Rightarrow\quad |A|=0

Property 5: Common factor from a row or column

If every element of one row or one column has a common factor k, it may be taken outside the determinant.

\begin{vmatrix}ka&kb\\c&d\end{vmatrix}=k\begin{vmatrix}a&b\\c&d\end{vmatrix}

Property 6: Common factor from every row or column

If a square determinant of order n has k multiplied into every element, the determinant is multiplied by kⁿ.

|kA|=k^n|A|

Property 7: Adding a multiple of one row to another

Adding k times one row to a different row does not change the determinant.

R_i\to R_i+kR_j\quad(i\neq j)\quad\Rightarrow\quad |A|\text{ unchanged}

The same property holds for columns.

Property 8: Row or column linearity

If a row or column is the sum of two corresponding rows or columns, the determinant can be split into two determinants.

D(R_i+S_i)=D(R_i)+D(S_i)

Property 9: Transpose

Taking the transpose does not change the determinant.

|A^T|=|A|

Property 10: Determinant of a product

|AB|=|A||B|

Property 11: Determinant of a power

|A^m|=|A|^m

Property 12: Determinant of an inverse

When A is invertible:

|A^{-1}|=\frac{1}{|A|}

Property 13: Determinant of a scalar multiple

For an n × n matrix:

|kA|=k^n|A|

Property 14: Triangular matrices

For an upper or lower triangular matrix, the determinant is the product of the diagonal entries.

\left|\begin{matrix}a&*&*\\0&b&*\\0&0&c\end{matrix}\right|=abc

Property 15: Identity matrix

|I_n|=1

Property 16: Singular matrix

|A|=0\quad\Longleftrightarrow\quad A\text{ is singular}

Property 17: Non-singular matrix

|A|\neq0\quad\Longleftrightarrow\quad A\text{ is non-singular}

10. Row and Column Operations — Quick Formula Table

Operation	Effect on determinant
Interchange two rows	Sign changes.
Interchange two columns	Sign changes.
Multiply one row by k	Determinant becomes k times the original.
Multiply one column by k	Determinant becomes k times the original.
Multiply every row of an n × n matrix by k	Determinant becomes kⁿ times the original.
Multiply every column by k	Determinant becomes kⁿ times the original.
R_i → R_i + kR_j	No change.
C_i → C_i + kC_j	No change.
Two rows equal/proportional	Determinant is zero.
Two columns equal/proportional	Determinant is zero.

11. Important Cofactor Identities

Expansion using elements and their corresponding cofactors gives:

a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}=|A|

If elements of one row are multiplied by cofactors belonging to a different row, the sum is zero:

a_{11}C_{21}+a_{12}C_{22}+a_{13}C_{23}=0

More generally:

\sum_{j=1}^{n}a_{ij}C_{kj}=\begin{cases}|A|,&i=k\\0,&i\neq k\end{cases}

12. Area of a Triangle Using Determinants

If the vertices of a triangle are , its area is:

\Delta=\frac12\left|\begin{matrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{matrix}\right|

The absolute value is used because area is non-negative.

Condition for collinearity

If the three points are collinear, their triangle has zero area:

\left|\begin{matrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{matrix}\right|=0

13. Equation of a Line Using a Determinant

The equation of the line passing through two points and can be written as:

\begin{vmatrix}x&y&1\\x_1&y_1&1\\x_2&y_2&1\end{vmatrix}=0

This determinant expresses that the three points are collinear.

14. Adjoint of a Matrix

The adjoint (adjugate) of a square matrix is the transpose of its cofactor matrix. If the cofactor matrix is C, then:

\operatorname{adj}(A)=C^T

For a 3 × 3 matrix:

\operatorname{adj}(A)=\begin{bmatrix}C_{11}&C_{21}&C_{31}\\C_{12}&C_{22}&C_{32}\\C_{13}&C_{23}&C_{33}\end{bmatrix}

For a 2 × 2 matrix:

\operatorname{adj}\begin{bmatrix}a&b\\c&d\end{bmatrix}=\begin{bmatrix}d&-b\\-c&a\end{bmatrix}

These adjoint relations are standard in the NCERT treatment of determinants and inverse matrices.

15. Fundamental Adjoint Identity

A\operatorname{adj}(A)=\operatorname{adj}(A)A=|A|I

This identity is the key step connecting determinants with the inverse of a matrix.

16. Determinant of the Adjoint

For a square matrix A of order n:

|\operatorname{adj}(A)|=|A|^{n-1}

For example, for a 3 × 3 matrix:

|\operatorname{adj}(A)|=|A|^2

17. Inverse of a Matrix Using Determinants

A square matrix has an inverse only when its determinant is non-zero. The formula is:

A^{-1}=\frac{1}{|A|}\operatorname{adj}(A),\qquad |A|\neq0

If , the inverse does not exist.

Important inverse identities

AA^{-1}=A^{-1}A=I

(AB)^{-1}=B^{-1}A^{-1}

(A^T)^{-1}=(A^{-1})^T

18. Solving Linear Equations Using Matrices

A system of linear equations can be written in matrix form as:

AX=B

|A|\neq0

then the unique solution is:

X=A^{-1}B

This is one of the important applications of determinants and matrices in algebra.

19. Cramer's Rule

For a system of two or three linear equations, determinants can also be used through Cramer's rule. For a two-variable system:

a_1x+b_1y=c_1,\qquad a_2x+b_2y=c_2

Define:

D=\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix}

D_x=\begin{vmatrix}c_1&b_1\\c_2&b_2\end{vmatrix},\qquad D_y=\begin{vmatrix}a_1&c_1\\a_2&c_2\end{vmatrix}

When D ≠ 0:

x=\frac{D_x}{D},\qquad y=\frac{D_y}{D}

This is a useful determinant-based method for solving small systems. For syllabus-specific preparation, always follow the method required by the relevant board or examination.

20. Useful Standard Determinant Identities

Some determinant forms frequently appear in higher-level algebra and competitive-exam problems.

Vandermonde determinant of order 3

\begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix}=(a-b)(b-c)(c-a)

Another common cyclic determinant

\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}=-a^3-b^3-c^3+3abc

Useful factorisation

a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)

Equivalent form

a^3+b^3+c^3-3abc=(a+b+c)\big((a-b)^2+(b-c)^2+(c-a)^2\big)/2

21. Determinants and Geometry

Determinants are not only algebraic objects. Their value can describe geometric scaling. In two dimensions, the absolute value of a determinant of a transformation matrix gives the area-scaling factor. In three dimensions, the absolute value gives a volume-scaling factor.

\text{Area scale}=|\det A|

\text{Volume scale}=|\det A|

For a general n-dimensional linear transformation, the absolute value of the determinant gives the corresponding n-dimensional volume scaling factor.

22. Complete Determinant Formula Sheet — Quick Revision

Concept	Formula / Result
Order 1
Order 2
Order 3
Minor	= delete row i and column j and take the determinant
Cofactor
Row expansion
Column expansion
Transpose
Product
Power
Scalar multiple
Inverse determinant
Adjoint
Adjoint identity
Determinant of adjoint
Inverse
Triangle area
Collinearity	Triangle determinant = 0
Line through two points
Singular matrix
Non-singular matrix
Linear system
Matrix solution	when

23. Common Mistakes in Determinants

Confusing a determinant with a matrix. A matrix is an array; a determinant is a number associated with a square matrix.
Forgetting the alternating + − + signs while expanding a 3 × 3 determinant.
Using the wrong minor by deleting the wrong row or column.
Forgetting that .
Assuming that adding a multiple of a row changes the determinant. It does not, provided it is added to a different row.
Forgetting that interchanging two rows changes the sign.
Using when .
Forgetting the absolute value when using a determinant to calculate geometric area.

24. Frequently Asked Questions

Is a determinant a matrix?

No. A matrix is an array of entries, whereas its determinant is a single number obtained from a square matrix.

Can a rectangular matrix have a determinant?

In the standard matrix-and-determinant framework, determinants are defined for square matrices.

What is the most important determinant formula?

For Class 12, the most frequently used core results include the 2 × 2 and 3 × 3 determinant formulas, minor and cofactor formulas, determinant properties, area formula, adjoint identity and inverse formula.

What does determinant zero mean?

It means the square matrix is singular. Consequently, its inverse does not exist.

Why are determinants useful beyond school mathematics?

They are part of linear algebra and are used in geometry, systems of equations, engineering, physics, computer science, optimisation and many mathematical models.

25. Final Takeaway

A determinant turns a square matrix into a single meaningful number. Learning its properties is often more important than memorising isolated tricks because the properties allow difficult-looking determinants to be simplified quickly.

Remember the core chain: Matrix → Determinant → Minors → Cofactors → Adjoint → Inverse → Linear equations.

For examination preparation, master the determinant properties first, then practise expansion, minors and cofactors, followed by area, adjoint, inverse and applications to equations.