Determinants | ISC Class 12 Maths Notes

This note covers determinants of square matrices, expansion by rows and columns, minors and cofactors, properties of determinants, triangle area and collinearity, adjoints, inverses, and the consistency and solution of linear systems in two or three variables.

What is a determinant, and how is its order identified?

A matrix is a rectangular arrangement of entries in horizontal rows and vertical columns. A square matrix has the same number of rows and columns. Its order is n × n, where n is that common number; its determinant is called a determinant of order n.

Definition: A determinant is the number associated with a square matrix. For a matrix called A, this number is written det A or |A|. Here |A| means the determinant of A, not the absolute value of a matrix.

We work with real entries and orders one, two and three. Real numbers include positive numbers, negative numbers and zero. A matrix and its determinant are different objects: the matrix is an arrangement, whereas its determinant is one number obtained by a specified calculation.

How should the notation be read?

Write A = [aᵢⱼ], where aᵢⱼ denotes the entry in row i and column j. The symbols i and j are row and column indices, respectively. Thus a₁₂ is in the first row and second column; the subscripts describe its position, not multiplication.

We use Rᵢ for row i and Cⱼ for column j. Square brackets enclose a matrix; vertical bars around an array denote its determinant. The notation det A is useful when vertical bars might otherwise be confused with absolute value.

For a one-entry matrix A = [a], where a is a real number, det A = a. The determinant can therefore be negative or zero. Only square matrices have determinants; counting both rows and columns comes before choosing an evaluation rule.

How is a determinant of order two evaluated?

For a matrix with rows (a, b) and (c, d), the letters a, b, c and d represent its four real entries. The main diagonal runs from the top left to the bottom right. Multiply its entries, then subtract the product of the other two entries.

$$\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc.$$

The subtraction has a fixed direction. Reversing it generally changes the sign. Negative entries must keep their signs inside brackets during substitution. In particular, subtracting a negative product increases the result; it is not equivalent to subtracting the magnitudes of both products.

Worked example 1. Evaluate the determinant with rows (2, 4) and (−1, 2).

Answer: The main-diagonal product is 2 × 2 = 4. The other product is 4 × (−1) = −4. Therefore the determinant is 4 − (−4) = 8.

How are algebraic entries handled?

The rule also applies when entries contain a variable, meaning a symbol representing an unspecified number. Evaluate the products algebraically before combining terms. A determinant containing a variable need not depend on that variable after simplification.

Worked example 2. Let x be a real variable. Evaluate the determinant with rows (x, x + 1) and (x − 1, x).

Answer: The determinant is x × x − (x + 1)(x − 1). Since (x + 1)(x − 1) = x² − 1, it becomes x² − (x² − 1) = 1.

The second example illustrates an identity, an equality that holds for every allowed value of its variable. Its result is independent of x. Substituting one value of x may check a calculation, but it does not establish an identity for all real x.

When equating two determinants, first replace each with its algebraic value. The equality is between numbers, so corresponding matrix entries need not be equal. Simplify the resulting equation and retain every real solution that satisfies it.

How do minors and cofactors control expansion?

Definition: The minor Mᵢⱼ of an entry aᵢⱼ is the determinant left after deleting row i and column j. Its cofactor Aᵢⱼ is the signed minor: Aᵢⱼ = (−1) raised to the power i + j, multiplied by Mᵢⱼ.

The capital letter in Aᵢⱼ denotes a cofactor, while the lower-case letter in aᵢⱼ denotes an entry. The minor of an entry in a determinant of order n, with n at least two, has order n − 1. Deleting a row and column preserves the remaining entries' order.

The cofactor sign is positive if i + j is even and negative if i + j is odd. This sign multiplies the minor's actual value. A negative minor in a negative-sign position therefore produces a positive cofactor.

Worked example 3. Find the minor of the entry 6 in the determinant with rows (1, 2, 3), (4, 5, 6) and (7, 8, 9).

Answer: The entry is in row 2, column 3. Delete that row and column. The remaining rows are (1, 2) and (7, 8), so M₂₃ = 1 × 8 − 2 × 7 = −6.

How does a minor differ from a cofactor?

For the two-row determinant with rows (1, −2) and (4, 3), the minors M₁₁, M₁₂, M₂₁ and M₂₂ are 3, 4, −2 and 1. Applying their positional signs gives cofactors A₁₁ = 3, A₁₂ = −4, A₂₁ = 2 and A₂₂ = 1.

Keep deletion and sign assignment as separate steps. The entry's own sign does not decide its cofactor sign. A cofactor is not found by multiplying the minor by the entry; that multiplication belongs to determinant expansion.

How is a determinant of order three expanded?

Result: Expansion along any row or column

A determinant equals the sum of the entries of any one row or column multiplied by their corresponding cofactors. Expansion means replacing the original determinant by this sum. All three rows and all three columns give the same value.

For a three-row matrix A, first-row expansion is det A = a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃. The alternating signs are already contained in the cofactors. When using minors instead, write a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃.

1. Choose one row or column, preferably one containing the greatest number of zeros.
2. For each chosen entry, delete its row and column to find the appropriate minor.
3. Apply the positional sign to obtain the cofactor, then multiply by the original entry.
4. Add the resulting terms and evaluate each remaining determinant of order two.

Worked example 4. Evaluate the determinant with rows (1, 2, 4), (−1, 3, 0) and (4, 1, 0).

Answer: Expand down column 3. Only the first entry contributes, and its positional sign is positive. The determinant is 4[(−1) × 1 − 3 × 4] = 4(−13) = −52. Both other terms vanish because their entries are zero.

What does a zero entry actually remove?

A zero entry makes its own entry-times-cofactor term zero. It does not imply that its minor or cofactor is zero. This distinction matters when calculating an adjoint, because the cofactors of zero entries must still be found.

For second-row expansion, the minor signs run negative, positive, negative. Do not restart the first row's pattern simply because the second row is now being used. The indices of each entry continue to determine the sign.

Note: If cofactors have already been calculated, add the entry-times-cofactor products directly. Applying the alternating signs a second time changes the calculation and can give an incorrect determinant.

Which properties simplify determinants?

Properties of determinants allow an array to be simplified before expansion. They apply to columns as well as rows. In the rules below, k denotes a real scalar, meaning a single number used to multiply entries. A transpose, written Aᵀ, interchanges a matrix's rows and columns.

Property: Transposition and interchange

det Aᵀ = det A: transposing the whole matrix leaves its determinant unchanged. Interchanging two rows, or two columns, changes the sign of the determinant. These operations are different: transposition exchanges every row with its corresponding column.

If two rows or columns are identical, the determinant is zero. More generally, if one is a scalar multiple of another, the determinant is zero. A row or column consisting entirely of zeros also makes the determinant zero.

Property: Scaling and row replacement

Multiplying every entry in one row by k multiplies the determinant by k. A common factor may therefore be taken outside from an entire row or column. Multiplying the whole matrix by k affects every row: det(kA) = kⁿ det A for order n.

Adding a multiple of one row to a different row leaves the determinant unchanged. The notation Rᵢ → Rᵢ + kRⱼ means replace row i with that sum, keeping row j unchanged, where i and j are different. The corresponding column operation works in the same way.

If each entry of one row is a sum of two terms, the determinant splits into two determinants, with the other rows unchanged. Only the selected row is split. This is addition in one row, not a general rule that the determinant of a sum of matrices equals a sum of determinants.

How can these properties be checked and used?

For order two, row replacement follows directly from the formula: replacing (a, b) by (a + kc, b + kd) gives (a + kc)d − (b + kd)c = ad − bc. The added products cancel.

For the determinant in worked example 3, replacing R₃ by R₃ − R₂ produces (3, 3, 3). Replacing R₂ by R₂ − R₁ then produces the same row. The determinant is therefore zero, because its second and third rows are identical.

Record any row interchange or factor extraction explicitly. Row replacement preserves the value, whereas interchange and scaling change it in stated ways. Tracking those differences is essential when using properties to create zeros before expansion.

How do determinants give triangle area and collinearity?

Let the three vertices, meaning corner points, of a triangle have coordinates (x₁, y₁), (x₂, y₂) and (x₃, y₃). Each x-coordinate gives horizontal position and each y-coordinate gives vertical position. Let D denote the determinant formed by placing each coordinate pair beside a final entry 1.

$$D=\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}.$$

The area is one half of the absolute value of D. Here absolute value, written |D|, means its non-negative magnitude. Thus area = ½|D|. In expanded form, D = x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂).

Worked example 5. Find the area of the triangle with vertices (3, 8), (−4, 2) and (5, 1).

Answer: D = 3(2 − 1) − 8(−4 − 5) + 1(−4 − 10) = 3 + 72 − 14 = 61. The area is 61/2 square units.

How are collinear points and unknown coordinates treated?

Collinear points lie on one straight line. Their triangle has zero area, so the coordinate determinant is zero. To find the equation of a line through two given points, use a general third point (x, y) on that line and equate the determinant to zero.

Worked example 6. Find the line joining P(1, 3) and Q(0, 0). Also find k if T(k, 0) makes triangle PQT have area 3 square units.

Answer: For a point (x, y) on the line, the determinant with rows (0, 0, 1), (1, 3, 1), (x, y, 1) is y − 3x. Hence y = 3x. For PQT, half the coordinate determinant is −3k/2, so −3k/2 = ±3 and k = ±2.

The symbol ± means that both the positive and negative alternatives must be considered. When an area is given, the determinant may have either sign. Solving just one signed equation can discard a valid position of the unknown vertex.

How is the adjoint of a square matrix formed?

Definition: The adjoint of A, denoted adj A, is the transpose of the matrix of its cofactors. First place each cofactor in the position of its corresponding entry, then interchange rows and columns.

The cofactor matrix has entries Aᵢⱼ. In adj A, the cofactor Aᵢⱼ occupies row j and column i. Thus the first row of the cofactor matrix becomes the first column of the adjoint. Forming the cofactor matrix alone does not complete the operation.

Worked example 7. Find the adjoint of the matrix A with rows (2, 3) and (1, 4).

Answer: Its cofactors are A₁₁ = 4, A₁₂ = −1, A₂₁ = −3 and A₂₂ = 2. The cofactor matrix has rows (4, −1), (−3, 2). Transposing gives adj A with rows (4, −3), (−1, 2).

For order two, this produces a shortcut: interchange the main-diagonal entries and change the signs of the other two entries. The shortcut follows from the cofactor definition. For order three, find the nine cofactors and transpose their arrangement.

Theorem: The adjoint identity

Let I denote the identity matrix, whose main-diagonal entries are 1 and whose other entries are 0. Then A(adj A) = (adj A)A = (det A)I. Juxtaposition here means matrix multiplication: each product entry is a row-times-column sum.

The diagonal entries of A(adj A) are ordinary cofactor expansions and equal det A. The other entries combine one row's entries with a different row's corresponding cofactors; each such sum is zero. This explains the identity's diagonal and zero entries.

In particular, a₁₁A₂₁ + a₁₂A₂₂ + a₁₃A₂₃ = 0 for order three. Do not confuse this with first-row expansion, which uses A₁₁, A₁₂ and A₁₃. The distinction between matching and different rows links determinant expansion to the adjoint identity.

When does an inverse exist, and how is it calculated?

A square matrix is singular when its determinant is zero and non-singular when its determinant is nonzero. The inverse of A, written A⁻¹, is a matrix satisfying AA⁻¹ = A⁻¹A = I. The exponent −1 here denotes a matrix inverse.

Theorem: Invertibility and the determinant

A square matrix has an inverse if and only if it is non-singular. When det A ≠ 0, the adjoint identity can be divided by that nonzero number to obtain A⁻¹ = (adj A)/(det A). Divide every entry of the adjoint by the determinant.

Conversely, if A has an inverse, taking determinants in AA⁻¹ = I gives (det A)(det A⁻¹) = 1. Thus det A cannot be zero. This uses the product rule det(AB) = (det A)(det B), where A and B are square matrices of the same order.

Worked example 8. Find the inverse of A with rows (1, 3, 3), (1, 4, 3), (1, 3, 4).

Answer: det A = 1(16 − 9) − 3(4 − 3) + 3(3 − 4) = 1. The cofactor rows are (7, −1, −1), (−3, 1, 0), (−3, 0, 1). Therefore adj A has rows (7, −3, −3), (−1, 1, 0), (−1, 0, 1). Since det A = 1, these are also the rows of A⁻¹.

How can an inverse be verified?

Multiply the proposed inverse by the original matrix and compare the result with I. For the example, multiplying A by its adjoint gives rows (1, 0, 0), (0, 1, 0), (0, 0, 1). This verifies both the cofactor calculation and the final division.

Check the determinant before calculating every cofactor if the only aim is to find an inverse. A zero determinant means that the inverse does not exist, even though the adjoint remains defined. Taking reciprocals of individual matrix entries is not the inverse operation.

How does the matrix method solve linear equations?

A linear equation combines unknowns to the first power with fixed numerical coefficients and a constant term. For a system, let A be the coefficient matrix, X the column of unknowns, and B the column of constants. Then the equations are represented together by AX = B.

In two variables x and y, X has entries x and y in that order. In three variables x, y and z, it has entries x, y and z. Each row of A must use that same order, including a zero coefficient wherever an unknown is absent.

Why is the inverse multiplied on the left?

If det A ≠ 0, multiply AX = B on the left by A⁻¹. Regrouping gives (A⁻¹A)X = A⁻¹B, so IX = A⁻¹B and X = A⁻¹B. This is the matrix method for a unique solution, meaning exactly one set of unknown values.

Worked example 9. Solve 2x + 5y = 1 and 3x + 2y = 7 by the matrix method.

Answer: A has rows (2, 5), (3, 2); B has entries 1, 7. Its determinant is 4 − 15 = −11. The inverse is −1/11 times the matrix with rows (2, −5), (−3, 2). Thus X = −1/11 times the column with entries −33, 11. Hence x = 3 and y = −1.

How does the procedure extend to three variables?

Worked example 10. Solve 3x − 2y + 3z = 8, 2x + y − z = 1 and 4x − 3y + 2z = 4.

Answer: A has rows (3, −2, 3), (2, 1, −1), (4, −3, 2), and det A = −17. Its adjoint has rows (−1, −5, −1), (−8, −6, 9), (−10, 1, 7). Multiplying this adjoint by B, the column with entries 8, 1, 4, gives entries −17, −34, −51. Dividing by −17 gives x = 1, y = 2, z = 3.

Finish by substituting the solution into every original equation. This checks the constants as well as the coefficient order. Obtaining a plausible value from one equation does not establish that it solves the whole system.

How is the consistency of a system decided?

A system is consistent if at least one solution exists. It is inconsistent if no solution exists. Consistency does not mean uniqueness: a system with infinitely many solutions is also consistent. For square coefficient matrices, a nonzero determinant guarantees a unique solution.

What changes when the determinant is zero?

Let O denote a zero matrix, whose entries are all zero, of the required size. If A is singular, premultiplying AX = B by adj A gives O = (adj A)B. Therefore, if (adj A)B ≠ O, the system is inconsistent.

Note: If det A = 0 and (adj A)B = O, the system may be either consistent or inconsistent. The zero product does not by itself prove that solutions exist. Further examination of the original equations is required.

How can inconsistency be demonstrated directly?

Consider x + 3y = 5 and 2x + 6y = 8. The second left-hand side is twice the first, but twice the first right-hand side would be 10. Subtracting twice the first equation from the second gives 0 = −2, a contradiction. No pair (x, y) can satisfy both.

When examining equations, a contradiction is a false numerical equality produced from them. An equation that merely repeats information does not create a contradiction. In a singular system, compatible equations can leave unknowns undetermined and allow infinitely many solutions.

Keep the logical order clear: test the determinant, use an inverse only when it exists, and interpret singular cases separately. The statement that the zero adjoint product “may” accompany either outcome must retain that uncertainty; it is not a sufficient consistency test.

Glossary

• Square matrix — A matrix with the same number of rows and columns, permitting a determinant to be defined.
• Determinant — A number associated with a square matrix through the specified rules of evaluation.
• Order — The common number of rows and columns in a square matrix or its determinant.
• Minor — The determinant remaining after deleting the row and column containing the selected entry.
• Cofactor — A minor multiplied by the positive or negative sign determined by the entry's position.
• Expansion — Evaluation by summing products of one row's or column's entries with their corresponding cofactors.
• Transpose — The matrix obtained by interchanging corresponding rows and columns of the original matrix.
• Adjoint — The transpose of the cofactor matrix associated with a given square matrix.
• Singular matrix — A square matrix whose determinant is zero and which therefore has no inverse.
• Non-singular matrix — A square matrix whose determinant is nonzero and which therefore possesses an inverse.
• Inverse — A matrix giving the identity matrix when multiplied by the original matrix in either order.
• Collinear points — Points lying on one straight line, giving zero area in the triangle determinant formula.
• Consistent system — A system of equations for which at least one common solution exists.
• Inconsistent system — A system of equations for which no common set of unknown values exists.

Common errors and misconceptions

• Misconception: A determinant and a matrix are the same object. Correct: A matrix is an arrangement of entries; its determinant is a number. Determinants are defined for square matrices.
• Misconception: A minor includes the cofactor sign. Correct: First calculate the minor by deletion, then multiply it by the positional sign to obtain the cofactor.
• Misconception: Every expansion starts with a positive minor term. Correct: Signs follow the original positions. Second-row expansion has the signs negative, positive, negative.
• Misconception: Multiplying a matrix by a scalar multiplies its determinant just once. Correct: Every row is scaled, so the determinant acquires the scalar raised to the matrix's order.
• Misconception: The cofactor matrix is already the adjoint. Correct: Transpose the cofactor matrix. Cofactor row positions become column positions in the adjoint.
• Misconception: A negative coordinate determinant gives a negative triangle area. Correct: Take half its absolute value. If the area is given, consider both determinant signs.
• Misconception: A zero determinant proves that a system has no solution. Correct: It rules out the inverse method; the singular system needs further examination for consistency.
• Misconception: The inverse is formed by taking entrywise reciprocals. Correct: Divide the adjoint by the nonzero determinant and verify using matrix multiplication.

Exam-style questions with model answers

Q1. Evaluate the determinant with rows (2, 4) and (−1, 2), showing the two products. [2 marks]

1. The main-diagonal product is 2 × 2 = 4, and the other product is 4 × (−1) = −4.
2. Subtracting the second product gives 4 − (−4) = 8, the required determinant.

Q2. For the determinant with rows (1, −2) and (4, 3), find all four minors and all four cofactors. [4 marks]

1. For the entry in row 1, column 1, deletion leaves 3. Hence M₁₁ = 3 and the positive positional sign gives A₁₁ = 3.
2. For row 1, column 2, deletion leaves 4. Thus M₁₂ = 4 and A₁₂ = −4.
3. For row 2, column 1, deletion leaves −2. Thus M₂₁ = −2 and A₂₁ = 2.
4. For row 2, column 2, deletion leaves 1. Thus M₂₂ = 1 and A₂₂ = 1.

Q3. Evaluate the determinant with rows (1, 2, 4), (−1, 3, 0), (4, 1, 0), expanding along the third column. [3 marks]

1. The third column contains 4, 0 and 0, so the last two entry-times-cofactor terms vanish. The first entry has a positive positional sign.
2. Deleting row 1 and column 3 leaves rows (−1, 3), (4, 1). Its determinant is (−1) × 1 − 3 × 4 = −13.
3. The required determinant is therefore 4 × (−13) = −52.

Q4. Using determinants, find the area of the triangle with vertices (3, 8), (−4, 2), (5, 1). [3 marks]

1. Form the coordinate determinant D with rows (3, 8, 1), (−4, 2, 1), (5, 1, 1). The area is one half of its absolute value.
2. Expanding gives D = 3(2 − 1) − 8(−4 − 5) + (−4 − 10) = 3 + 72 − 14 = 61.
3. Therefore the area is ½|61| = 61/2 square units, a positive quantity.

Q5. Find the inverse of the matrix A with rows (1, 3, 3), (1, 4, 3), (1, 3, 4), using cofactors. [5 marks]

1. Expand along the first row: det A = 1(16 − 9) − 3(4 − 3) + 3(3 − 4) = 1. The matrix is non-singular, so its inverse exists.
2. Deleting the appropriate rows and columns and applying positional signs gives the first cofactor row (7, −1, −1).
3. The second and third cofactor rows are (−3, 1, 0) and (−3, 0, 1), respectively.
4. Transpose the cofactor matrix. The adjoint has rows (7, −3, −3), (−1, 1, 0), (−1, 0, 1).
5. Use A⁻¹ = (adj A)/(det A). Division by 1 leaves the adjoint unchanged, giving those same three rows for the inverse.

Q6. Solve 2x + 5y = 1 and 3x + 2y = 7 using the inverse matrix method, and verify the solution. [5 marks]

1. Write AX = B, with A having rows (2, 5), (3, 2); X the column of x, y; and B the column of 1, 7.
2. Calculate det A = 2 × 2 − 5 × 3 = −11. It is nonzero, so a unique solution exists.
3. The adjoint has rows (2, −5), (−3, 2). Therefore A⁻¹ is −1/11 times this matrix.
4. Multiply X = A⁻¹B. The intermediate column has entries 2 − 35 = −33 and −3 + 14 = 11. Hence x = 3 and y = −1.
5. Substitution gives 2(3) + 5(−1) = 1 and 3(3) + 2(−1) = 7, verifying both original equations.

Q7. Examine whether x + 3y = 5 and 2x + 6y = 8 are consistent. Explain why a zero coefficient determinant alone would not settle the question. [3 marks]

1. The coefficient determinant is 1 × 6 − 3 × 2 = 0, so the coefficient matrix has no inverse.
2. Twice the first equation requires 2x + 6y = 10, whereas the second requires 2x + 6y = 8. These conditions contradict each other.
3. The system is inconsistent. In general, a zero determinant alone does not establish inconsistency; a singular system may instead have infinitely many solutions.

Key takeaways

• A determinant is a number associated with a square matrix; begin by identifying its order and the positions of its entries.
• Calculate a minor by deletion and a cofactor by applying its positional sign before using either in an expansion.
• Expansion along any row or column gives the same determinant; choosing a row or column with zeros reduces arithmetic.
• Row replacement preserves a determinant, row interchange reverses its sign, and scaling one row scales the determinant once.
• Triangle area is half the absolute coordinate determinant, while collinearity requires that determinant to equal zero.
• The adjoint is the transpose of the cofactor matrix; dividing it by a nonzero determinant gives the inverse.
• A nonzero coefficient determinant guarantees a unique solution obtained by multiplying the constant column by the inverse on the left.
• For a singular coefficient matrix, a zero adjoint-times-constant product leaves consistency undecided and requires examination of the original equations.

Test yourself

Why does |A| not mean the absolute value of a matrix here?

The bars denote the determinant of the square matrix A, a number that may be positive, negative or zero.

What must be deleted to obtain M₂₃?

Delete row 2 and column 3, keeping the surviving entries in their original order, then evaluate the remaining determinant.

Why can the cofactor of a negative minor be positive?

A negative positional sign multiplies the minor by −1, so a negative minor in that position produces a positive cofactor.

How does multiplying every entry of a matrix of order three by k affect its determinant?

All three rows are multiplied by k, so the determinant is multiplied by k³, where k is the scalar used.

What extra operation turns a cofactor matrix into an adjoint?

Transpose it: each cofactor row becomes the corresponding column, and each cofactor column becomes the corresponding row.

Why are both signs considered when a triangle's area is given?

Area uses the absolute value of the coordinate determinant, so either sign of that determinant can give the same area.

When can AX = B be solved as X = A⁻¹B?

This formula applies when the square coefficient matrix A is non-singular, so its determinant is nonzero and its inverse exists.

What follows if det A = 0 and (adj A)B = O?

The system may be consistent or inconsistent. Examine the original equations; the zero product alone does not settle consistency.