Determinants Chapter 4 — Class 12 Mathematics Prepared by Bushra CK - HSST Mathematics, MJHSS Elettil

Chapter Overview - 4.1 Introduction — why determinants matter - 4.2 Determinants of order 1, 2, and 3×3 with expansion methods - 4.3 Area of a triangle using determinants - 4.4 Minors and Cofactors - 4.5 Adjoint and Inverse of a matrix - 4.6 Solving systems of linear equations (Matrix Method)

4.1 Why Determinants? Consider the system of two linear equations: - This system has a unique solution if and only if - The coefficient matrix is - The number is called the determinant of A - Notation: , , or

What is a Determinant? - Every square matrix has a unique number associated with it called its determinant - Think of it as a function where - Important: is read as "determinant of A", not modulus of A - Only square matrices have determinants - Determinants have applications in Engineering, Science, Economics, Social Science

4.2.1 Determinant of Order 1 For a matrix : Example: If , then If , then > Note: The determinant can be negative. It is NOT the same as absolute value.

4.2.2 Determinant of Order 2 For : Memory trick:

Ex 4.2 : Evaluate a 2×2 Determinant Evaluate: Solution:

Example 2 (4.2) — Determinant with Variables Evaluate: Solution: > The determinant is constant — independent of !

Exercise 4.1 — Problems 1 & 2 (Try These) Q1. Evaluate: Q2(i). Evaluate: Q2(ii). Evaluate:

Exercise 4.1 — Problems 1 & 2 (Solutions) Q1. Q2(i). Q2(ii).

Exercise 4.1 — Problems 3 & 4 (Scalar Multiple Rule) Q3. If , show that Key Rule: If and both are , then Here , so ✓ Q4. If , show that Here , so ✓

4.2.3 Determinant of a 3×3 Matrix — The Idea - Express as a combination of 2×2 determinants - Called expansion of a determinant along a row (or column) - There are 6 ways to expand — along or - All give the same value

Expansion Along Row 1 — Step by Step Take each element of with its sign and multiply by the 2×2 determinant formed by deleting that element's row and column: Sign pattern for :

Expansion — Full Formula Key tips: - Expand along the row/column that has the most zeros — fewer terms to compute - Instead of , simply use or based on whether is even or odd - Expansion along , , , , all give the same answer

Sign Pattern for 3×3 Expansion The sign for each position: - Position : - Position : - Position : - Position :

Example 3 (4.2) — Smart Choice of Column Evaluate: Strategy: has two zeros — expand along

Example 4 (4.2) — Trigonometric Determinant Evaluate: Expand along (first element is 0):

Example 5 (4.2) — Finding Unknown Find values of such that: Solution: LHS , RHS

Exercise 4.1 — Problems 5 & 6 Q5(i). Evaluate: (Hint: Expand along — two zeros) Q5(ii). Evaluate: Q6. Find where

Exercise 4.1 — Problems 5 & 6 (Solutions) Q5(i). Expand along : Q5(ii). Expanding along : Q6. > means the matrix is singular!

Exercise 4.1 — Problems 7 & 8 Q7(i). Find if: Q7(ii). Find if: Q8. If , then equals: (A) 6 (B) ±6 (C) −6 (D) 0

Exercise 4.1 — Problems 7 & 8 (Solutions) Q7(i). LHS ; RHS Q7(ii). LHS ; RHS Q8. LHS ; RHS

4.3 Area of a Triangle The area of a triangle with vertices , , is: Three key points: - Area is positive — always take the absolute value - If area is given, the determinant can be or (two cases) - If the three points are collinear, area

Example 6 (4.3) — Area of a Triangle Find the area of triangle with vertices , , . Expanding along :

Example 7 (4.3) — Line Equation + Area Problem Find the equation of line through and . Also find if gives area of sq. units. Part 1 — Equation of line AB: Let be any point on . Then area of :

Example 7 (4.3) — Continued Part 2 — Finding : Area of sq. units, so: So or .

Exercise 4.2 — Problems 1 & 2 Q1. Find area of triangle with vertices: - (i) , , - (ii) , , - (iii) , , Q2. Show that points , , are collinear. (Hint: Show that the area determinant equals zero)

Exercise 4.2 — Problem 1 (Solutions) Q1(i). sq. units Q1(ii). sq. units Q1(iii). 0 > Q1(iii): Area is 0, so the three points are collinear!

Exercise 4.2 — Problems 3, 4 & 5 Q3. Find if area sq. units with vertices: - (i) , , - (ii) , , Q4(i). Find equation of line joining and using determinants. Q4(ii). Find equation of line joining and using determinants. Q5. If area of triangle is 35 sq. units with vertices , and , then is: (A) 12 (B) −2 (C) −12, −2 (D) 12, −2

Exercise 4.2 — Problems 3, 4, 5 (Solutions) Q3(i). or Q3(ii). or Q4(i). Q4(ii). , i.e. Q5. Setting up: gives or → Answer: (D)

4.4 Minors — Definition Minor of element is the determinant obtained by deleting the -th row and -th column. - Minor of an element in an matrix has order Example 8 (4.4): In , find minor of 6. Element 6 is at position . Delete and :

4.4 Cofactors — Definition Cofactor of element is: Relation to determinant: This works for any row or column. Important Zero Property: If elements of one row are multiplied with cofactors of a different row, the sum is always zero.

Example 9 (4.4) — Minors and Cofactors of 2×2 Find all minors and cofactors of | Element | Minor | Cofactor | |---|---|---| | | | | | | | | | | | | | | | |

Example 10 (4.4) — Minors and Cofactors of 3×3 Find , , , for

Example 11 (4.4) — Full Cofactor Table Find all minors and cofactors of | | Minor | Cofactor | |---|---|---| | | | | | | | | | | | | | | | | | | | | | | | |

Example 11 (4.4) — Verification Continuing from the cofactor table: , ; , ; , Verify that : > This confirms: elements of × cofactors of = 0

Exercise 4.3 — Problems 1 & 2 Q1. Write minors and cofactors of all elements: - (i) - (ii) Q2(i). Q2(ii).

Exercise 4.3 — Problem 3 Q3. Using cofactors of elements of the second row, evaluate: Solution: Expand along :

Exercise 4.3 — Problems 4 & 5 Q4. Using cofactors of elements of the third column, evaluate: (Answer: ) Q5. If and are cofactors of , the value of is given by: (A) (B) (C) (D) Answer: (D) — sum of column 1 elements with their own cofactors.

4.5 Adjoint of a Matrix Definition: The adjoint of a square matrix is the transpose of the cofactor matrix: For a matrix, if cofactor matrix is , then:

Adjoint — 2×2 Shortcut For : Rule: Swap and ; negate and . Example 12 (4.5): Find adj for

Theorem 1 — A × adj A For any square matrix of order : Also: - Singular matrix: - Non-singular matrix: > Theorems 2 & 3: If , are non-singular, is non-singular;

4.5 Inverse of a Matrix Theorem 4: is invertible if and only if , and: Properties: - - - - Inverse is unique

Example 13 (4.5) — Verify A·adj A = |A|I, Find A⁻¹ Given , verify and find . Cofactors:

Example 13 — Continued Since :

Example 14 (4.5) — Verify Given , , verify . , ; ,

Example 15 (4.5) — Using Cayley-Hamilton to Find Inverse Show satisfies . Hence find . From : post-multiply by :

Exercise 4.4 — Problems 1 & 2 Q1. Find adj for Q2. Find adj for Q3. Verify for Q4. Verify for

Exercise 4.4 — Problems 5 to 11 (Find Inverse) Find the inverse of each matrix (if it exists): Q5. Q6. Q7. Q8. Q9. Q10. Q11.

Exercise 4.4 — Problems 12–16 Q12. Let , . Verify . Q13. If , show . Hence find . Q14. For , find , such that . Q15. For , show . Find . Q16. For , verify . Find .

Exercise 4.4 — MCQ Problems 17 & 18 Q17. Let be a nonsingular square matrix of order . Then equals: (A) (B) (C) (D) Answer: (B) — using with Q18. If is an invertible matrix of order 2, then equals: (A) (B) (C) (D) Answer: (B) — since , so

4.6 System of Equations — Matrix Form Consider: Write as where: If : premultiply by to get

Three Cases for AX = B | Condition | Meaning | Outcome | |---|---|---| | | Non-singular | Unique solution: | | , | Singular | Inconsistent — no solution | | , | Singular | Infinitely many or no solutions | - Consistent system: solution exists (one or more) - Inconsistent system: no solution exists - This chapter restricts to unique solution cases only

Example 16 (4.6) — Solving 2×2 System Solve: and , so unique solution exists.

Example 17 (4.6) — Solving 3×3 System Solve: , , Cofactors:

Example 17 — Continued

Example 18 (4.6) — Word Problem to Matrix Equation Sum of three numbers is 6. Multiply third by 3 and add second: get 11. First plus third equals double of second. Find the numbers. Setting up equations from conditions:

Example 18 — Continued Cofactors of :

Exercise 4.5 — Consistency Check (Q1–Q6) Examine consistency of each system: Q1. , Q2. , Q3. , Q4. , , Q5. , , Q6. , ,

Exercise 4.5 — Consistency Solutions (Q1–Q3) Q1. , → Consistent (unique solution) Q2. , → Consistent (unique solution) Q3. , → Inconsistent — no solution!

Exercise 4.5 — Solve Using Matrix Method (Q7–Q9) Q7. , Q8. , Q9. , Solutions: Q7. , → Q8. , Q9. ,

Exercise 4.5 — Q10 to Q12 Q10. , Q11. , , Q12. , , Solutions: Q10. Q11. ... (set up , find , compute ) Q12. ... (set up and solve)

Exercise 4.5 — Q13 & Q14 Q13. Solve using matrix method: Solution: , after computing and : Q14. Solve using matrix method: Solution: , solving gives

Exercise 4.5 — Q15 & Q16 Q15. If , find . Using , solve: (Answer: ) Q16. Cost of 4 kg onion, 3 kg wheat, 2 kg rice is ₹60. Cost of 2 kg onion, 4 kg wheat, 6 kg rice is ₹90. Cost of 6 kg onion, 2 kg wheat, 3 kg rice is ₹70. Find cost per kg of each item using matrix method. (Answer: Onion ₹5/kg, Wheat ₹8/kg, Rice ₹8/kg)

Miscellaneous Exercise — Q1 & Q2 Q1. Prove that the determinant is independent of . (Expand and show result is — no ) Q2. Evaluate: (Answer: 1 — expand along )

Miscellaneous Exercise — Q5 & Q6 Q5. Evaluate: (Answer: ) Q6. Evaluate: (Answer: )

Miscellaneous Exercise — MCQ Q8 & Q9 Q8. If are nonzero reals, the inverse of is: (A) (B) (C) (D) Answer: (A) — diagonal matrix inverse is inverse of each diagonal element.

Miscellaneous Exercise — Q9 Q9. Let , where . Then: Expand : - When : - When : - So Answer: (D)

Chapter Summary | Concept | Key Formula | |---|---| | Order 2 determinant | | | Area of triangle | | | Cofactor | | | Adjoint | Transpose of cofactor matrix | | Inverse | | | Matrix equation | when |

Key Takeaways - Determinant is a unique number; only square matrices have one; it can be negative - Expand along the row/column with most zeros to minimise computation - Collinear points → area determinant ; use this to test collinearity - exists if and only if (non-singular) - For : if → unique solution - Elements of row × cofactors of row always equals zero — crucial property

JEE Main 2019 — Maximum Value of a Determinant Function [JEE Main 2019] Find the maximum value of - (A) - (B) - (C) - (D)

JEE Main 2019 — Solution Step 1: Apply . Each entry in becomes (rows 1 & 2) and (row 3): Step 2: Apply , killing and entries in row 1: Step 3: Expand along : Step 4: Maximum of is : Answer: (B) — the key insight is reducing the trig determinant to a single trig expression via row operations.

JEE Advanced 2016 — Counting Distinct Real Roots [JEE Advanced 2016, Paper 1 — Integer Answer Type] Find the total number of distinct for which

JEE Advanced 2016 — Solution Step 1: Factor from and from — total factor of comes out: Step 2: Apply , to create zeros in : Step 3: Expand along : Step 4: Let : Each value of gives exactly one real . Answer: 2