Determinants — Study Notes

1. Introduction

A determinant is a scalar value that can be computed from the entries of a square matrix. Whereas a matrix is merely an array of numbers, its determinant condenses the whole matrix into a single number that encodes crucial information: whether the matrix is invertible, whether the associated system of linear equations has a unique solution, the area or volume of parallelograms and parallelepipeds, and much more.

This chapter defines determinants for square matrices of orders 1, 2, and 3, introduces the concepts of minors and cofactors, and develops the property-based expansion of determinants. It then derives the properties of determinants, the adjoint of a matrix, and the inverse of a matrix using the adjoint and the determinant. The chapter concludes with the application of determinants and matrices to solving systems of linear equations through Cramer's rule and the matrix method, together with conditions for consistency of a system.

The applications of determinants extend far beyond linear algebra. Determinants compute areas of triangles from coordinates, decide collinearity, and give the condition for a system of equations to have infinitely many solutions or no solution. The area interpretation, in particular, links determinants to geometry and will reappear in the chapter on application of integrals. Mastering the expansion techniques and the properties is the key to performing well in this high-weightage chapter.

2. Determinant of a Square Matrix

To every square matrix A, we associate a number called the determinant, written as det(A), |A|, or with the matrix enclosed in vertical bars.

Order 1

If $A = [a]$, then $|A| = a$.

Order 2

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

Order 3

$$|A| = \begin{vmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{vmatrix}$$

Expanding along the first row:

$$|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$$

Note the alternating signs: the sign for position (i, j) is $(-1)^{i+j}$.

3. Minors and Cofactors

The minor $M_{ij}$ of an element $a_{ij}$ is the determinant obtained by deleting the i-th row and j-th column of A. For a 3 x 3 matrix, each minor is a 2 x 2 determinant.

The cofactor $A_{ij}$ of $a_{ij}$ is defined as

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

The signs $(-1)^{i+j}$ form a checkerboard pattern, beginning with + in the top-left corner.

Expansion using Cofactors

The determinant can be expanded along any row or any column using cofactors:

$$|A| = \sum_{j=1}^{n} a_{ij} A_{ij} \quad \text{(expansion along row i)}$$

$$|A| = \sum_{i=1}^{n} a_{ij} A_{ij} \quad \text{(expansion along column j)}$$

4. Properties of Determinants

1. The value of a determinant does not change when rows and columns are interchanged: $|A^T| = |A|$.
2. Interchanging any two rows (or columns) changes the sign of the determinant.
3. If any two rows (or columns) of a determinant are identical or proportional, the determinant is zero.
4. A common factor of all elements of a row (or column) can be taken outside the determinant.
5. If each element of a row (or column) is a sum of two terms, the determinant can be expressed as the sum of two determinants.
6. The value of a determinant is unchanged if to the elements of a row (or column) we add a constant multiple of the corresponding elements of another row (or column): $R_i \to R_i + kR_j$.
7. For two square matrices of the same order, $|AB| = |A| \cdot |B|$.

These properties are used to simplify a determinant before expanding, ideally creating zeros in a row or column.

5. Area of a Triangle

The area of a triangle whose vertices are $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$ is given by

$$\text{Area} = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{vmatrix}$$

Since area is always taken as positive, the absolute value of the determinant is used. The points are collinear if and only if the determinant equals zero.

6. Adjoint and Inverse of a Matrix

Adjoint

The adjoint of a square matrix A, written adj(A), is the transpose of the matrix of cofactors. If C is the matrix of cofactors, then

$$\text{adj}(A) = C^T$$

Inverse

A square matrix A is invertible if and only if $|A| \neq 0$. The inverse is

$$A^{-1} = \frac{1}{|A|} \text{adj}(A)$$

For a 2 x 2 Matrix

If $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, then

$$\text{adj}(A) = \begin{bmatrix} d & -b \ -c & a \end{bmatrix}, \quad A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$$

Note that for a 2 x 2 matrix the adjoint is obtained by swapping the diagonal entries and changing the sign of the off-diagonal entries.

Key Identities

• $A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| I$
• $|A^{-1}| = \frac{1}{|A|}$
• $|\text{adj}(A)| = |A|^{n-1}$ for a matrix of order n
• A matrix is singular if $|A| = 0$ and non-singular if $|A| \neq 0$.

7. Solving Systems of Linear Equations

Matrix Method

The system $a_1 x + b_1 y + c_1 z = d_1$, $a_2 x + b_2 y + c_2 z = d_2$, $a_3 x + b_3 y + c_3 z = d_3$ can be written as AX = B, where

$$A = \begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} d_1 \ d_2 \ d_3 \end{bmatrix}$$

If $|A| \neq 0$, the system is consistent with the unique solution $X = A^{-1}B$.

Conditions for Consistency

• If $|A| \neq 0$: unique solution; the system is consistent.
• If $|A| = 0$ and adj(A)B = 0: infinitely many solutions; consistent.
• If $|A| = 0$ and adj(A)B $\neq$ 0: no solution; inconsistent.

Cramer's Rule

For the same system, if $|A| = D \neq 0$, then

$$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}$$

where $D_x$ is obtained by replacing the first column of D with the constants, $D_y$ by replacing the second column, and $D_z$ by replacing the third column.

Quick Revision Tables

Table 1: Determinant Formulas

Table 2: Properties of Determinants at a Glance

Table 3: Invertibility and Adjoint Quick Facts

Mind Map

Important Diagrams (SVG)

Diagram 1: Expansion of a 3 x 3 Determinant Using Minors

Diagram 2: Triangle Area via Determinant and Collinearity

Common Mistakes

1. Writing the determinant of a 2 x 2 matrix as ad + bc. The correct value is ad - bc.
2. Forgetting the alternating signs when expanding a 3 x 3 determinant along a row: the middle term always carries a minus sign.
3. Confusing a matrix with its determinant. A matrix is an array; a determinant is a single number, written with vertical bars instead of brackets.
4. Computing $A^{-1}$ without checking $|A| \neq 0$; if the determinant is zero, the matrix is singular and the inverse does not exist.
5. Using adj(A) for a 2 x 2 matrix as the cofactor matrix directly. The adjoint is the transpose of the cofactor matrix; for 2 x 2 this works out to swap diagonal and negate off-diagonal entries.
6. Forgetting the absolute value when computing the area of a triangle; the determinant may be negative and area must be positive.
7. Declaring a system inconsistent just because |A| = 0; it may have infinitely many solutions. Check adj(A)B.
8. Interchanging two rows and claiming the determinant is unchanged; the sign changes.
9. Taking out a common factor from only one element instead of the entire row or column.
10. In matrix method solutions, multiplying B A^(-1) instead of A^(-1) B. Since matrix multiplication is not commutative, the order matters: X = A^(-1)B.

Exam Tips

1. Before expanding a 3 x 3 determinant, use row operations to create zeros in one row or column, then expand along it. This saves significant computation time.
2. When checking collinearity, write the points as rows (x, y, 1) and compute the determinant; zero means collinear.
3. To find the inverse, first compute |A|. If it is zero, state that the inverse does not exist; otherwise use A^(-1) = adj(A)/|A|.
4. For 2 x 2 inverses, memorise the pattern: swap the main diagonal, negate the off-diagonal, divide by the determinant.
5. In MCQ questions on properties, test with a simple example such as swapping two rows of a 2 x 2 determinant to check sign changes.
6. For systems of equations, always compute |A| first; if it is non-zero there is a unique solution, and you can use either the matrix method or Cramer's rule.
7. Remember |AB| = |A||B|, and that |kA| = k^n |A| for an n x n matrix A; students often forget the power.
8. In word problems that reduce to systems of equations, write the system in matrix form AX = B clearly before solving; this earns method marks.

Conclusion

Determinants distill the essential properties of square matrices into a single number and provide the computational engine for linear algebra. The expansion by minors and cofactors, supported by the powerful simplification properties, gives efficient methods for evaluating determinants of any order. The area of a triangle and the collinearity condition connect determinants to coordinate geometry, while the adjoint and the inverse make the determinant the key to solving systems of linear equations. The consistency conditions for a system, whether unique solution, infinitely many, or none, hinge entirely on the value of the determinant and the behaviour of adj(A)B. Because the board examination frequently combines determinant properties with matrix inversion and equation solving, this chapter rewards careful methodical work more than any other algebra chapter in the syllabus.