How to Use This Page
This page provides complete exercise solutions for Chapter 1: Matrices and Determinants from the Samacheer Kalvi 11th Business Mathematics textbook. Each section follows the textbook exercise structure:
1. Read the Key Concepts section for a quick refresher before attempting problems. 2. Work through each exercise section — try the problem first, then check your answer. 3. Pay special attention to the worked examples in Section VI, as they demonstrate the calculation steps examiners expect. 4. Review the Common Mistakes section to avoid frequent errors students make in exams. 5. Use the Exam Checklist before your exam to verify you have covered every topic.
Note: In code blocks throughout this page, matrices are written in row format. For example, [[a,b],[c,d]] represents a 2×2 matrix with first row [a, b] and second row [c, d].
Key Concepts (Quick Recap)
Matrix Types
| Type | Definition | Example |
|---|---|---|
| Row Matrix | One row, n columns (1 × n) | [[2, 3, 5]] |
| Column Matrix | m rows, one column (m × 1) | [[1],[4],[7]] |
| Square Matrix | Same number of rows and columns (n × n) | [[1,2],[3,4]] |
| Diagonal Matrix | Square matrix with non-zero entries only on the main diagonal | [[3,0],[0,7]] |
| Scalar Matrix | Diagonal matrix where all diagonal entries are equal | [[5,0],[0,5]] |
| Identity (Unit) Matrix | Diagonal matrix with all diagonal entries = 1; denoted I | [[1,0],[0,1]] |
| Zero (Null) Matrix | All entries are zero; denoted O | [[0,0],[0,0]] |
| Symmetric Matrix | A = Aᵀ (equals its transpose) | [[1,3],[3,5]] |
| Skew-Symmetric Matrix | A = −Aᵀ (negative of its transpose); diagonal entries are 0 | [[0,2],[-2,0]] |
Matrix Operations
| Operation | Rule | Condition |
|---|---|---|
| Addition | A + B = add corresponding entries | Same order (both m × n) |
| Subtraction | A − B = subtract corresponding entries | Same order (both m × n) |
| Scalar Multiplication | kA = multiply every entry by scalar k | Any matrix |
| Matrix Multiplication | AB = row of A × column of B (dot product) | Columns of A = Rows of B |
| Transpose | Aᵀ = swap rows and columns | Any matrix |
Key multiplication rule: If A is m × n and B is p × q, then AB exists only if n = p, and the result is m × q.
Transpose Properties
Determinant of a 2×2 Matrix
Determinant of a 3×3 Matrix (Rule of Sarrus / Expansion)
Properties of Determinants
| Property | Statement | ||||||
|---|---|---|---|---|---|---|---|
| Value unchanged | Aᵀ | = | A | ||||
| Row swap | Swapping two rows changes the sign: | A' | = − | A | |||
| Identical rows | If two rows (or columns) are identical, | A | = 0 | ||||
| Scalar multiplication | If each element of a row is multiplied by k, | A' | = k | A | |||
| Triangular matrix | A | = product of diagonal elements | |||||
| Factor property | If each element is a sum, determinant splits into sum of determinants | ||||||
| Product rule | AB | = | A | × | B | ||
| Singular | A | = 0 means A is singular (no inverse) | |||||
| Non-singular | A | ≠ 0 means A is non-singular (inverse exists) |
Adjoint and Inverse
Cramer's Rule
Solving Systems of Linear Equations
Method 1 — Matrix Inversion: If AX = B, then X = A⁻¹B.
| Method 2 — Cramer's Rule: xᵢ = | Aᵢ | / | A | , where Aᵢ is A with column i replaced by B. |
|---|
Method 3 — Row Reduction (Gauss elimination): Transform the augmented matrix to row echelon form.
For the TN Board Business Maths exam, Methods 1 and 2 are most frequently tested.
I. Choose the Best Answer (MCQs)
[[5, 0], [0, 5]] is:| **4. If ` | A | = 4 and A is a 2 × 2 matrix, then | 2A | ` is:** |
|---|
(a) 4   (b) 8   (c) 16   (d) 2
[[1, 0], [0, 1]] is:II. Fill in the Blanks
1. A matrix with only one row is called a ______ matrix.
A row matrix has dimension 1 × n, meaning one row and n columns.
2. If A = [[2, −1], [3, 4]], then | A | = ______. |
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|A| = (2)(4) − (−1)(3) = 8 + 3 = 11
3. Two matrices can be added only if they are of the same ______.
Matrix addition is defined only for matrices of the same order (same number of rows and columns).
4. The adjoint of [[3, 1], [2, 5]] is ______.
For a 2×2 matrix [[a,b],[c,d]], adj(A) = [[d,−b],[−c,a]] = [[5,−1],[−2,3]].
5. A square matrix A is singular if ______.
| Answer: | A | = 0 |
|---|
A matrix is called singular when its determinant is zero. A singular matrix does not have an inverse.
| 6. If A is a 3 × 3 matrix and | A | = 5, then | 3A | = ______. |
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For a 3 × 3 matrix, |kA| = k³|A|. So |3A| = 3³ × 5 = 27 × 5 = 135.
7. The transpose of a column matrix is a ______ matrix.
A column matrix (m × 1) becomes a row matrix (1 × m) upon transposition.
8. If A is symmetric, then Aᵀ = ______.
By definition, a symmetric matrix satisfies Aᵀ = A.
9. The determinant of an identity matrix of order 3 is ______.
The determinant of any identity matrix, regardless of order, is 1.
| 10. In Cramer's rule, x = | A₁ | / | A | , where A₁ is obtained by replacing column 1 of A with ______. |
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A₁ is formed by replacing the first column of the coefficient matrix A with the column of constants B from AX = B.
| 11. If ` | A | = 0`, then A⁻¹ ______. |
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The inverse formula A⁻¹ = adj(A)/|A| requires |A| ≠ 0. Division by zero is undefined, so no inverse exists for a singular matrix.
12. The product of a matrix and its inverse gives the ______ matrix.
AA⁻¹ = A⁻¹A = I, where I is the identity matrix of the same order.
III. True or False
1. Every square matrix has an inverse.
Correction: Only non-singular square matrices (those with |A| ≠ 0) have inverses. A singular matrix (|A| = 0) has no inverse.
2. Matrix multiplication is commutative, i.e., AB = BA for all square matrices A and B.
Correction: Matrix multiplication is generally not commutative. AB ≠ BA in most cases. For example, if A = [[1,1],[0,0]] and B = [[0,0],[1,1]], then AB = [[1,1],[0,0]] but BA = [[0,0],[1,1]].
3. The determinant of a diagonal matrix is the product of its diagonal elements.
For a diagonal matrix, all off-diagonal elements are zero, so the determinant simplifies to the product of the diagonal elements.
4. If two rows of a determinant are interchanged, the value of the determinant remains the same.
Correction: Interchanging two rows changes the sign of the determinant. If the original value is D, after swapping two rows it becomes −D.
5. A skew-symmetric matrix of odd order is always singular.
For a skew-symmetric matrix, Aᵀ = −A. Taking determinants: |Aᵀ| = |−A|. Since |Aᵀ| = |A| and for odd order n, |−A| = (−1)ⁿ|A| = −|A|, we get |A| = −|A|, which gives |A| = 0.
IV. Match the Following
Match Column A with the correct answer from Column B.
| # | Column A | # | Column B | ||
|---|---|---|---|---|---|
| 1 | Identity matrix | A | A | = 0 | |
| 2 | Singular matrix | B | A = Aᵀ | ||
| 3 | Symmetric matrix | C | Diagonal entries = 1, rest = 0 | ||
| 4 | Skew-symmetric matrix | D | A | ≠ 0 | |
| 5 | Non-singular matrix | E | Aᵀ = −A | ||
| 6 | Scalar matrix | F | All diagonal entries equal, rest = 0 |
1 → C (Identity matrix has diagonal entries = 1 and all other entries = 0)
2 → A (Singular matrix has determinant = 0)
3 → B (Symmetric matrix satisfies A = Aᵀ)
4 → E (Skew-symmetric matrix satisfies Aᵀ = −A)
| 5 → D (Non-singular matrix has | A | ≠ 0) |
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6 → F (Scalar matrix has equal diagonal entries and zeros elsewhere)
V. Answer Briefly
VI. Answer in Detail
1. Matrix Multiplication and Its Properties (with Worked Example)
Properties of Matrix Multiplication:
1. Non-commutative: In general, AB ≠ BA. 2. Associative: (AB)C = A(BC), provided the products are defined. 3. Distributive: A(B + C) = AB + AC and (A + B)C = AC + BC. 4. Identity: AI = IA = A (where I is the identity matrix of appropriate order). 5. Null product: AO = OA = O (where O is the zero matrix).
7. Transpose of product: (AB)ᵀ = BᵀAᵀ.
| 6. Determinant of product: | AB | = | A | × | B | . |
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Worked Example:
Let A = [[2, 3], [1, 4]] and B = [[5, 0], [2, 1]]. Find AB and BA, and verify that AB ≠ BA.
| **Verification of determinant property | AB | = | A | × | B | :** |
|---|
2. Computing Determinant and Inverse of a 2×2 Matrix (with Worked Example)
Steps to find the inverse:
5. Verify: AA⁻¹ should equal I.
| 1. Compute the determinant | A | . | ||
|---|---|---|---|---|
| 2. If | A | = 0, the inverse does not exist (A is singular). | ||
| 3. If | A | ≠ 0, find the adjoint: swap the diagonal elements and negate the off-diagonal elements. | ||
| 4. Multiply the adjoint by 1/ | A | : A⁻¹ = (1/ | A | ) × adj(A). |
Worked Example:
Find the inverse of A = [[3, 2], [7, 5]] and verify your answer.
3. Solving a System of Linear Equations Using Cramer's Rule (with Worked Example)
Cramer's Rule Statement:
For the system:
Worked Example:
Solve the system using Cramer's rule:
Verification:
3×3 System Extension (Cramer's Rule):
Common Mistakes to Avoid
1. Assuming AB = BA: Matrix multiplication is not commutative. Always multiply in the given order. Unless specifically told A and B commute, never assume AB = BA.
2. Forgetting dimension compatibility: Before multiplying A (m × n) and B (p × q), verify n = p. The result is m × q. Attempting to multiply incompatible matrices is a common exam error.
3. Confusing determinant with matrix: A determinant is a single number (scalar), while a matrix is an array. You cannot "add a determinant to a matrix" or treat them interchangeably.
4. Wrong adjoint formula: For [[a,b],[c,d]], the adjoint is [[d,−b],[−c,a]] — swap the diagonal entries (a↔d) and negate the off-diagonal entries (b→−b, c→−c). Do not just negate all entries.
| 5. Dividing by zero determinant: The formula A⁻¹ = adj(A)/ | A | requires | A | ≠ 0. Always compute | A | first and check it is non-zero before finding the inverse. |
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| 6. Wrong Cramer's rule substitution: In | A₁ | , replace only column 1 with B, keeping all other columns exactly as in A. Similarly for | A₂ | , replace only column 2. Do not shuffle multiple columns. |
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| 7. Scalar multiplication vs. determinant scaling: For an n × n matrix A, | kA | = kⁿ | A | , not k | A | . The scalar multiplies every entry, so the factor appears n times (once per row) in the determinant. |
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8. Sign errors in 3×3 determinant expansion: When expanding along the first row, the signs alternate: +a₁M₁ − a₂M₂ + a₃M₃. Students often forget the alternating signs or get the cofactor pattern wrong (+, −, + for row 1).
Exam Checklist
Use this checklist before your exam to ensure you have covered all topics:
- [ ] I can define and identify all matrix types (row, column, square, diagonal, scalar, identity, zero, symmetric, skew-symmetric).
- [ ] I can add, subtract, and multiply matrices, and I know the dimension rules for each operation.
- [ ] I can compute the transpose of a matrix and state its properties (especially (AB)ᵀ = BᵀAᵀ).
- [ ] I can calculate the determinant of 2×2 and 3×3 matrices correctly.
- [ ] I can state and apply all key properties of determinants (row swap sign change, identical rows → zero, triangular → diagonal product, etc.).
- [ ] I can find the adjoint of a 2×2 matrix using the correct formula.
- [ ] I can find the inverse of a 2×2 matrix using A⁻¹ = adj(A)/|A| and verify by checking AA⁻¹ = I.
- [ ] I can solve a system of 2 linear equations in 2 variables using Cramer's rule.
- [ ] I can solve a system of linear equations using the matrix inversion method (X = A⁻¹B).
- [ ] I can distinguish between singular (|A| = 0) and non-singular (|A| ≠ 0) matrices and explain what this means for solvability.