Determinants Complete Revision: Formulas, Properties, Cramer’s Rule

Determinants carry long calculations, many signs and several properties, so students often lose marks through small slips. Knowing which property to apply can cut a question down to a few lines.

These Determinants revision notes bring together minors, cofactors, properties, special determinants and Cramer's rule in one clear order. For better preparation, explore PW’s Class 12 Maths resources to practise questions and strengthen your understanding of key concepts.

What is a Determinant and How is It Different from a Matrix?

A matrix is an arrangement of numbers in rows and columns. Its order is written as number of rows x number of columns, so a matrix with 2 rows and 2 columns has order 2 x 2. A square matrix has an equal number of rows and columns, and a determinant can be found only for a square matrix.

The determinant of A is written as det(A) or with vertical lines. For a second-order determinant:

| a b ; c d | = ad - bc

The table shows the main differences between a matrix and a determinant.

How Do You Find Minors and Cofactors?

For an element A(ij) in a determinant, i gives the row number and j gives the column number. The minor M(ij) is found by deleting row i and column j. The cofactor is:

C(ij) = (-1)^(i+j) x M(ij)

For a third-order determinant, the cofactor signs follow this pattern:

| + - + ; - + - ; + - + |

For example, C12 = -M12 and C22 = +M22.

Memory tip: To find a minor, delete the element's row and column. To find its cofactor, multiply the minor by (-1)^(i+j).

How Do You Expand a Determinant and Use Sarrus' Rule?

A third-order determinant can be expanded along any row or column. For D = | a b c ; d e f ; g h i |, expansion along the first row gives:

D = a x M11 - b x M12 + c x M13

Using cofactors, this is D = a x C11 + b x C12 + c x C13. Expansion along the second row follows the sign pattern -, +, -.

Sarrus' rule works only for a determinant of order three. Repeat the first two columns on the right of the determinant. Then:

• Add the products of the three downward diagonals.

• Add the products of the three upward diagonals.

• Subtract the upward-diagonal sum from the downward-diagonal sum.

Sarrus' rule is useful when sine and cosine terms appear, because mixed terms may cancel using sin^2(theta) + cos^2(theta) = 1.

What Are the Important Properties of Determinants?

Properties help you simplify a determinant before expanding it. The table lists the results you should know by heart.

For an n x n matrix, det(kA) = k^n x det(A). This differs from multiplying only one row by k, which multiplies the determinant by k.

Which Row and Column Operations Keep the Determinant Unchanged?

The operations R(i) to R(i) + lambda x R(j) and C(i) to C(i) + lambda x C(j) do not change the determinant. The row or column being replaced must keep coefficient 1.

An operation such as R(i) to 2R(i) - R(j) changes the determinant, because the original row is multiplied by 2. To simplify large determinants, create zeros and reduce the number of non-zero entries.

What are the Cofactor Identities?

For a square matrix A, elements multiplied by cofactors from the same row or column give the determinant:

a11 x C11 + a12 x C12 + a13 x C13 = det(A)

If elements from one row are multiplied by cofactors from another row, the sum is zero:

a11 x C21 + a12 x C22 + a13 x C23 = 0

For an n x n matrix, if C is the cofactor matrix:

det(C) = [det(A)]^(n-1)

For a third-order matrix, det(C) = [det(A)]^2.

What Are the Special Determinants?

Some determinants have fixed results that save time in exams. The Vandermonde-type determinant is:

| 1 1 1 ; a b c ; a^2 b^2 c^2 | = (a - b)(b - c)(c - a)

This is useful when numbers are in arithmetic progression or when the determinant has first and second powers. Other standard results are:

• | 1 1 1 ; a b c ; bc ca ab | = (a - b)(b - c)(c - a)

• | 1 1 1 ; a b c ; a^3 b^3 c^3 | = (a - b)(b - c)(c - a)(a + b + c)

• | a b c ; b c a ; c a b | = -(a^3 + b^3 + c^3 - 3abc)

Memory tip: Learn these special values, as they appear in standard and proof-based questions.

For positive a, b and c, the expression (a - b)^2 + (b - c)^2 + (c - a)^2 helps decide whether a determinant expression is positive or negative.

How Do You Differentiate, Sum or Integrate a Determinant?

When variables occur in a determinant, differentiate one row at a time while keeping the other rows unchanged. The derivative is the sum of the determinants formed by differentiating each row separately.

For summation or integration:

• If the variable is in only one row or column, apply the operation directly to that row or column.

• If the variable is in more than one row or column, first evaluate the determinant and then apply the summation or integration.

Useful sums: the sum of 1 from k = 1 to n is n, the sum of 2k is n(n + 1), and the sum of (2k - 1) is n^2.

Memory tip: One row or column means apply the operation directly. More than one means determinant first, operation second.

How Do You Find the Product of Two Determinants?

When each element of a determinant has two or three terms, it may be the product of two smaller determinants. The usual method has these steps:

1. Count the terms in each element.

2. Identify the likely order of the two determinants.

3. Rebuild their row or column patterns.

4. Verify the product through row-by-row multiplication.

5. Apply determinant identities or row operations.

Row-by-row multiplication is recommended because it makes factorisation easier to spot.

What Is Cramer's Rule for Systems of Equations?

Cramer's rule solves a system of linear equations using determinants. For the system a1x + b1y + c1z = d1, a2x + b2y + c2z = d2 and a3x + b3y + c3z = d3, let Delta be the determinant of the coefficients:

Delta = | a1 b1 c1 ; a2 b2 c2 ; a3 b3 c3 |

Find Delta x, Delta y and Delta z by replacing the matching coefficient column with the constants d1, d2, d3. Then:

x = Delta x / Delta, y = Delta y / Delta, z = Delta z / Delta

The table gives the three possible outcomes. A system is consistent if it has at least one solution.

What Happens in a Homogeneous System?

In a homogeneous system, all constants are zero. If Delta is not 0, the only solution is the trivial one, x = y = z = 0. If Delta = 0, non-trivial solutions may exist.

Geometrically, three equations represent three planes. Coincident planes may give infinitely many solutions, while distinct parallel planes give no solution.

What Are the Applications of Determinants?

Determinants are used to find the area of a triangle and the equation of a line through two points. They also give the condition for three concurrent lines, where the determinant of their coefficients is set equal to zero. 

They are used in conditions related to the general second-degree equation and in properties of conics such as the parabola, ellipse, hyperbola and circle.

Determinants Important Questions

Solving a few well-chosen questions shows how well you have understood the properties and formulas. The questions here cover common exam patterns, with short answers for quick revision.

1. Can a determinant be found for a 2 x 3 matrix? 

2. If A is a 3 x 3 matrix with det(A) = 4, find det(2A). 

3. If A is a 3 x 3 matrix with det(A) = 4, find det(A^3). 

4. Find the value of | 1 1 1 ; 1 2 3 ; 1 4 9 |. 

5. Solve x + y = 3 and x - y = 1 using Cramer's rule. 

6. What is the determinant of an odd-order skew-symmetric matrix? 

7. What does Delta = 0 tell us about a system of equations?

8. If Δ1 = | x sinθ cosθ ; −sinθ −x 1 ; cosθ 1 x | and Δ2 = | x sin2θ cos2θ ; −sin2θ −x 1 ; cos2θ 1 x |, then for all θ ∈ (0, π/2):

A. Δ1 − Δ2 = x(cos2θ − cos4θ)
B. Δ1 + Δ2 = −2x³
C. Δ1 + Δ2 = −2(x³ + x − 1)
D. Δ1 − Δ2 = −2x³

9. Let A = [aij] = [ log₅128 log₄5 ; log₅8 log₄25 ]. If Aij is the cofactor of aij, Cij = Σ (k = 1 to 2) aik·Ajk, 1 ≤ i, j ≤ 2, and C = [Cij], then 8|C| is equal to:

A. 242
B. 288
C. 262
D. 222

10. Let M and m respectively be the maximum and the minimum values of f(x) = | 1 + sin²x cos²x 4sin4x ; sin²x 1 + cos²x 4sin4x ; sin²x cos²x 1 + 4sin4x |, x ∈ R. Then M⁴ − m⁴ is equal to:

A. 1295
B. [not visible in the image]
C. 1215
D. 1280

11. Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)^(i + j − 2) · aij, where i, j = 1, 2, 3. If the determinant of B is 81, then the determinant of A is:

A. 1/3
B. 3
C. 1/81
D. 1/9

12. Let the numbers 2, b, c be in an A.P. and A = [ 1 1 1 ; 2 b c ; 4 b² c² ]. If det(A) ∈ [2, 16], then c lies in the interval:

A. (2 + 2^(3/4), 4)
B. [4, 6]
C. [3, 2 + 2^(3/4))

Key Tips to Revise Determinants

Short daily practice works better than long, rare sessions for this chapter. The tips here help you avoid the common mistakes in Determinants.

• Check that the matrix is square before finding a determinant.

• Learn the sign pattern + - + for cofactors and use (-1)^(i+j) every time.

• Remember det(kA) = k^n x det(A), and do not confuse it with multiplying one row by k.

• Use row and column operations to create zeros before expanding.

• Avoid operations like R(i) to 2R(i) - R(j) unless you adjust for the factor of 2.

• Use Sarrus' rule only for third-order determinants.

• Write all special determinant results on one formula sheet and revise it often. 

• In Cramer's rule, find Delta first and then decide which case applies.

Regular revision of determinant formulas, properties and Cramer's rule can help you solve questions faster and avoid calculation errors. Practise different question types and revise key results regularly to build confidence for Class 12 and JEE exams.

 PW's IIT JEE preparation and Class 12 online courses offer concept classes and practice sets that can support your revision.

Related Resources

• Class 12 Maths Notes

• Maths Formulas

• Class 11 Maths Notes

• IIT JEE

• Class 12 Online Course