Determinants

Determinant: Definition and Notation

A **determinant** is a function that associates a unique number (real or complex) with every square matrix. For a square matrix A, the determinant is denoted by |A|, det(A), or ∆.

**Important Remarks:**

Only **square matrices** have determinants

|A| is read as "determinant of A," not "modulus of A"

Determinants are used to determine the uniqueness of solutions to systems of linear equations, find areas of triangles, and verify invertibility of matrices

For a 2×2 matrix A = [a b; c d], the determinant is calculated as |A| = ad - bc

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Determinant of a Matrix of Order One

For a matrix A = [a] of order 1×1, the determinant is simply:

**det(A) = a**

This is the foundational case from which all higher-order determinants are built.

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Determinant of a Matrix of Order Two

For a matrix A = [a₁₁ a₁₂; a₂₁ a₂₂] of order 2×2:

**|A| = a₁₁a₂₂ - a₂₁a₁₂**

**Worked Example 1:** Evaluate |2 4; -1 2|

|A| = 2(2) - 4(-1) = 4 + 4 = **8**

**Worked Example 2:** Evaluate |x x+1; x-1 x|

|A| = x(x) - (x+1)(x-1) = x² - (x² - 1) = x² - x² + 1 = **1**

**Key Point:** The order of subtraction matters—always compute the product of diagonal elements (top-left to bottom-right) minus the product of off-diagonal elements (top-right to bottom-left).

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Determinant of a Matrix of Order Three

For a 3×3 matrix A = [aᵢⱼ], the determinant can be expanded along any row or column. All six possible expansions (three rows + three columns) yield the same result.

Expansion Along First Row (R₁)

**|A| = (-1)^(1+1) a₁₁ M₁₁ + (-1)^(1+2) a₁₂ M₁₂ + (-1)^(1+3) a₁₃ M₁₃**

Or in expanded form:

**|A| = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)**

where Mᵢⱼ is the minor (2×2 determinant obtained by deleting row i and column j).

Expansion Along Any Row or Column

The determinant can be expanded along:

**Row 1, Row 2, or Row 3:** Use the same pattern with alternating signs

**Column 1, Column 2, or Column 3:** Same principle—multiply each element by (-1)^(i+j) times its minor

**General Rule:** When expanding along a row/column, the sign pattern follows (-1)^(i+j) where i is the row and j is the column of the element.

**Strategy for Calculation:** Expand along the row or column containing the **maximum number of zeros** to minimize computational effort.

Worked Example 3: Evaluate ∆ = |1 2 4; -1 3 0; 4 1 0|

Since column 3 has two zeros, expand along C₃:

∆ = 4(-1)^(1+3) |(-1) 3; 4 1| + 0 + 0

∆ = 4(1)[(-1)(1) - (3)(4)]

∆ = 4[-1 - 12] = 4(-13) = **-52**

Worked Example 4: Evaluate ∆ = |0 sin α -cos α; -sin α 0 sin β; cos α -sin β 0|

Expand along R₁:

∆ = 0 - sin α((-sin α)(0) - (sin β)(cos α)) - cos α((−sin α)(−sin β) - 0)

∆ = sin α sin β cos α - cos α sin α sin β = **0**

Important Remark on Scalar Multiplication

If A = kB where A and B are square matrices of order n:

**|A| = kⁿ|B|**

Example: If A = 2B where both are 2×2 matrices, then |A| = 2²|B| = 4|B|

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Area of a Triangle Using Determinants

For a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃), the area is:

**Area = (1/2)|x₁ y₁ 1; x₂ y₂ 1; x₃ y₃ 1|**

**Key Points:**

Always take the **absolute value** of the determinant since area is positive

If the determinant equals zero, the three points are **collinear** (form no triangle)

When area is given, use both positive and negative values: Area = ±(1/2)|determinant|

Worked Example 5: Find area of triangle with vertices (3, 8), (-4, 2), (5, 1)

Area = (1/2)|3 8 1; -4 2 1; 5 1 1|

Expanding along R₁:

= (1/2)|3(2-1) - 8(-4-5) + 1(-4-10)|

= (1/2)|3 - 8(-9) + 1(-14)|

= (1/2)|3 + 72 - 14| = (1/2)|61| = **30.5 square units**

Worked Example 6: Equation of a line and collinearity condition

For points A(1, 3), B(0, 0), and any point P(x, y) on line AB:

|x y 1; 1 3 1; 0 0 1| = 0 (area of triangle ABP is zero)

Expanding: (1/2)[3x - y] = 0 gives **y = 3x** (equation of line AB)

For a third point D(k, 0) such that area of triangle ABD = 3 sq units:

|1 3 1; 0 0 1; k 0 1| = ±6

This gives 3k/2 = ±3, so **k = ±2**

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Minors and Cofactors

Definition of Minor

The **minor Mᵢⱼ** of element aᵢⱼ is the determinant obtained by deleting the iᵗʰ row and jᵗʰ column from the original determinant.

Minor of an element in an n×n determinant is an (n-1)×(n-1) determinant

Minors are always numerical values (not matrices)

Definition of Cofactor

The **cofactor Aᵢⱼ** of element aᵢⱼ is defined as:

**Aᵢⱼ = (-1)^(i+j) Mᵢⱼ**

The sign (-1)^(i+j) follows a checkerboard pattern: positive for (i+j) even, negative for (i+j) odd.

Worked Example 7: Find minors and cofactors of |1 -2; 4 3|

**Minors:**

M₁₁ = 3

M₁₂ = 4

M₂₁ = -2

M₂₂ = 1

**Cofactors:**

A₁₁ = (-1)^(1+1)(3) = 3

A₁₂ = (-1)^(1+2)(4) = -4

A₂₁ = (-1)^(2+1)(-2) = 2

A₂₂ = (-1)^(2+2)(1) = 1

Expansion Using Cofactors

Any determinant can be expressed as:

**|A| = sum of (element × cofactor) for any single row or column**

For expansion along R₁: |A| = a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃

For expansion along C₁: |A| = a₁₁A₁₁ + a₂₁A₂₁ + a₃₁A₃₁

**Critical Property:** If elements of one row (or column) are multiplied by cofactors of a **different** row (or column), the sum equals zero:

a₁₁A₂₁ + a₁₂A₂₂ + a₁₃A₂₃ = 0

This property is essential for proving properties of adjoint matrices.

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Adjoint of a Matrix

Definition of Adjoint

The **adjoint of matrix A** (denoted adj A) is the transpose of the cofactor matrix:

**If C = [Aᵢⱼ] is the cofactor matrix, then adj A = Cᵀ**

For a 3×3 matrix:

First, find all nine cofactors Aᵢⱼ

Arrange them in a matrix C

Transpose C to get adj A

For a 2×2 matrix A = [a b; c d], there is a shortcut:

**adj A = [d -b; -c a]** (interchange diagonals, negate off-diagonals)

Worked Example 8: Find adj A for A = |1 2; 3 4|

Using the shortcut formula:

**adj A = |4 -2; -3 1|**

Verification:

A₁₁ = 4, A₁₂ = -3, A₂₁ = -2, A₂₂ = 1

Cofactor matrix = |4 -3; -2 1|

adj A = transpose = |4 -2; -3 1| ✓

Fundamental Theorem of Adjoint

**A(adj A) = (adj A)A = |A|I**

where I is the identity matrix and |A|I means |A| multiplied by each entry of I.

**Proof Sketch:** Using the properties of minors and cofactors—when we multiply a row of A by cofactors from the same row, we get |A|; when we multiply by cofactors from a different row, we get 0. The product yields a diagonal matrix with |A| on all diagonal entries.

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Singular and Non-Singular Matrices

Definitions

A square matrix A is **singular** if **|A| = 0**

A square matrix A is **non-singular** if **|A| ≠ 0**

**Example of Singular Matrix:** A = |1 2; 4 8|

|A| = 1(8) - 2(4) = 0 (singular)

**Example of Non-Singular Matrix:** A = |1 2; 3 4|

|A| = 1(4) - 2(3) = -2 ≠ 0 (non-singular)

**Significance:**

Only **non-singular matrices are invertible**

Singular matrices represent degenerate transformations (e.g., all vectors map to a line or plane)

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Inverse of a Matrix

Condition for Invertibility

**A square matrix A is invertible if and only if A is non-singular** (i.e., |A| ≠ 0)

Formula for Inverse

If A is invertible:

**A⁻¹ = (1/|A|) × adj A**

**Verification Theorem:** If A is invertible, then:

**AA⁻¹ = A⁻¹A = I**

Worked Example 9: Find A⁻¹ for A = |1 3; 3 1|

Step 1: Calculate |A| = 1(1) - 3(3) = 1 - 9 = -8 ≠ 0 (invertible)

Step 2: Find adj A = |1 -3; -3 1|

Step 3: Apply formula

**A⁻¹ = (1/-8)|1 -3; -3 1| = |-1/8 3/8; 3/8 -1/8|**

Properties of Inverse (for non-singular matrices)

**(AB)⁻¹ = B⁻¹A⁻¹** (order reverses)

**(A⁻¹)⁻¹ = A**

**(Aᵀ)⁻¹ = (A⁻¹)ᵀ**

**|A⁻¹| = 1/|A|**

Worked Example 10: Verify (AB)⁻¹ = B⁻¹A⁻¹

Given A = |2 -3; 1 2| and B = |-1 4; 1 -3|

Calculate: AB = |-5 17; 1 -2|, |AB| = 10-17 = -7

(AB)⁻¹ = (1/-7)|−2 −17; -1 -5| = |2/7 17/7; 1/7 5/7|

Also compute separately:

A⁻¹ = (1/7)|2 3; -1 2|

B⁻¹ = (1/-1)|−3 −4; -1 -1| = |3 4; 1 1|

B⁻¹A⁻¹ = (1/7)|2 17; 1 5| = |2/7 17/7; 1/7 5/7| ✓

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Important Determinant Properties

**Property 1 - Transpose:** |Aᵀ| = |A|

**Property 2 - Scalar Multiplication:** |kA| = kⁿ|A| for n×n matrix

**Property 3 - Row/Column Swap:** Swapping two rows (or columns) changes sign of determinant

**Property 4 - Identical Rows/Columns:** If two rows or columns are identical, |A| = 0

**Property 5 - Row Operation:** Adding a multiple of one row to another does not change the determinant

**Property 6 - Product of Matrices:** |AB| = |A||B|

**Property 7 - Adjoint Determinant:** |adj A| = |A|ⁿ⁻¹ for n×n matrix

**Exam Strategy:** Use these properties to simplify calculations before expanding—multiply rows/columns by scalars, perform row operations to create zeros, then expand.

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Summary of Key Formulas

**2×2 Determinant:** |a b; c d| = ad - bc

**3×3 by Expansion:** |A| = Σ(aᵢⱼ × cofactor Aᵢⱼ) along any row/column

**Area of Triangle:** Area = (1/2)|determinant of coordinate matrix|

**Adjoint:** adj A is transpose of cofactor matrix

**Inverse:** A⁻¹ = (adj A)/|A| if |A| ≠ 0

**Determinant Property:** A(adj A) = |A|I

**Scalar Multiple:** |kA| = kⁿ|A|

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Common Exam Mistakes to Avoid

**Confusing minor and cofactor:** Minor is just a determinant; cofactor includes the sign factor (-1)^(i+j)

**Wrong expansion formula:** Remember sign pattern—NOT all positive

**Forgetting absolute value for area:** Area must always be positive

**Assuming all matrices are invertible:** Only non-singular matrices (|A| ≠ 0) have inverses

**Wrong order in (AB)⁻¹:** Always reverse: B⁻¹A⁻¹, not A⁻¹B⁻¹

**Expanding along wrong row/column:** Always choose the row/column with most zeros

**Arithmetic errors:** Double-check sign calculations in cofactors—this is where most mistakes occur