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Matrices & Determinants Solutions | KalviGavani

Samacheer Kalvi 11th Business Maths Matrices and Determinants exercise solutions — matrix operations, types, determinants, inverse, Cramer's rule, solving linear equations, worked examples, MCQs, full answers.

Updated 15 June 202620 min read

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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

TypeDefinitionExample
Row MatrixOne row, n columns (1 × n)[[2, 3, 5]]
Column Matrixm rows, one column (m × 1)[[1],[4],[7]]
Square MatrixSame number of rows and columns (n × n)[[1,2],[3,4]]
Diagonal MatrixSquare matrix with non-zero entries only on the main diagonal[[3,0],[0,7]]
Scalar MatrixDiagonal matrix where all diagonal entries are equal[[5,0],[0,5]]
Identity (Unit) MatrixDiagonal matrix with all diagonal entries = 1; denoted I[[1,0],[0,1]]
Zero (Null) MatrixAll entries are zero; denoted O[[0,0],[0,0]]
Symmetric MatrixA = Aᵀ (equals its transpose)[[1,3],[3,5]]
Skew-Symmetric MatrixA = −Aᵀ (negative of its transpose); diagonal entries are 0[[0,2],[-2,0]]

Matrix Operations

OperationRuleCondition
AdditionA + B = add corresponding entriesSame order (both m × n)
SubtractionA − B = subtract corresponding entriesSame order (both m × n)
Scalar MultiplicationkA = multiply every entry by scalar kAny matrix
Matrix MultiplicationAB = row of A × column of B (dot product)Columns of A = Rows of B
TransposeAᵀ = swap rows and columnsAny 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

1. (Aᵀ)ᵀ = A (Double transpose returns original) 2. (A + B)ᵀ = Aᵀ + Bᵀ (Transpose distributes over addition) 3. (kA)ᵀ = kAᵀ (Scalar factor comes out unchanged) 4. (AB)ᵀ = BᵀAᵀ (Transpose reverses multiplication order)

Determinant of a 2×2 Matrix

For A = [[a, b], [c, d]] |A| = ad − bc Example: A = [[3, 4], [1, 2]] |A| = (3)(2) − (4)(1) = 6 − 4 = 2

Determinant of a 3×3 Matrix (Rule of Sarrus / Expansion)

For A = [[a₁, a₂, a₃], [b₁, b₂, b₃], [c₁, c₂, c₃]] |A| = a₁(b₂c₃ − b₃c₂) − a₂(b₁c₃ − b₃c₁) + a₃(b₁c₂ − b₂c₁) (expand along first row using cofactors)

Properties of Determinants

PropertyStatement
Value unchangedAᵀ=A
Row swapSwapping two rows changes the sign:A'= −A
Identical rowsIf two rows (or columns) are identical,A= 0
Scalar multiplicationIf each element of a row is multiplied by k,A'= kA
Triangular matrixA= product of diagonal elements
Factor propertyIf each element is a sum, determinant splits into sum of determinants
Product ruleAB=A×B
SingularA= 0 means A is singular (no inverse)
Non-singularA≠ 0 means A is non-singular (inverse exists)

Adjoint and Inverse

Adjoint of a 2×2 matrix: For A = [[a, b], [c, d]] adj(A) = [[d, −b], [−c, a]] Inverse: A⁻¹ = (1/|A|) × adj(A) [exists only if |A| ≠ 0] For the 2×2 case: A⁻¹ = (1/(ad − bc)) × [[d, −b], [−c, a]]

Cramer's Rule

For the system: a₁x + b₁y = c₁ a₂x + b₂y = c₂ Coefficient matrix: A = [[a₁, b₁], [a₂, b₂]] |A| = a₁b₂ − a₂b₁ |A₁| = [[c₁, b₁], [c₂, b₂]] → |A₁| = c₁b₂ − c₂b₁ |A₂| = [[a₁, c₁], [a₂, c₂]] → |A₂| = a₁c₂ − a₂c₁ x = |A₁|/|A|, y = |A₂|/|A| (Requires |A| ≠ 0)

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)

1. If A is a 2 × 3 matrix and B is a 3 × 4 matrix, then the order of AB is:
Answer

(a) 2 × 4   (b) 3 × 3   (c) 2 × 3   (d) 3 × 4

Answer(a) 2 × 4
ExplanationFor A (m × n) = 2 × 3 and B (n × p) = 3 × 4, AB has order m × p = 2 × 4. The inner dimensions (3 and 3) must match for multiplication to be valid.
2. The determinant of the matrix [[5, 0], [0, 5]] is:
Answer

(a) 0   (b) 5   (c) 25   (d) 10

Answer(c) 25
Explanation|A| = (5)(5) − (0)(0) = 25. This is a scalar matrix with diagonal entries 5.
3. If A is a square matrix such that Aᵀ = −A, then A is:
Answer

(a) Symmetric   (b) Skew-symmetric   (c) Diagonal   (d) Scalar

Answer(b) Skew-symmetric
ExplanationBy definition, a matrix is skew-symmetric when Aᵀ = −A. The diagonal elements of a skew-symmetric matrix are always 0.
**4. If `A= 4 and A is a 2 × 2 matrix, then 2A` is:**

(a) 4   (b) 8   (c) 16   (d) 2

Answer(c) 16
ExplanationFor a 2 × 2 matrix, multiplying every element by 2 multiplies the determinant by 2² = 4. So |2A| = 4 × 4 = 16. In general, for an n × n matrix, |kA| = kⁿ|A|.
5. The inverse of [[1, 0], [0, 1]] is:
Answer

(a) [[0, 1], [1, 0]]   (b) [[−1, 0], [0, −1]]   (c) [[1, 0], [0, 1]]   (d) Does not exist

Answer(c) [[1, 0], [0, 1]]
ExplanationThis is the identity matrix I. Since I⁻¹ = I, the inverse of the identity matrix is itself. Also |I| = 1 ≠ 0, so the inverse exists.
6. If A and B are square matrices of the same order, then (AB)ᵀ equals:
Answer

(a) AᵀBᵀ   (b) BᵀAᵀ   (c) AB   (d) BA

Answer(b) BᵀAᵀ
ExplanationThis is the transpose reversal property. The transpose of a product reverses the order: (AB)ᵀ = BᵀAᵀ.
7. If the system AX = B has a unique solution, then:
Answer
(a)A= 0   (b)A≠ 0   (c)A= 1   (d) A = I
**Answer: (b)A≠ 0**
ExplanationA unique solution exists only when the coefficient matrix is non-singular, i.e., |A| ≠ 0. If |A| = 0, the system has either no solution or infinitely many solutions.

II. Fill in the Blanks

1. A matrix with only one row is called a ______ matrix.

Answerrow

A row matrix has dimension 1 × n, meaning one row and n columns.

2. If A = [[2, −1], [3, 4]], thenA= ______.
Answer11

|A| = (2)(4) − (−1)(3) = 8 + 3 = 11

3. Two matrices can be added only if they are of the same ______.

Answerorder

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 ______.

Answer[[5, −1], [−2, 3]]

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 andA= 5, then3A= ______.
Answer135

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.

Answerrow

A column matrix (m × 1) becomes a row matrix (1 × m) upon transposition.

8. If A is symmetric, then Aᵀ = ______.

AnswerA

By definition, a symmetric matrix satisfies Aᵀ = A.

9. The determinant of an identity matrix of order 3 is ______.

Answer1

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 ______.
Answerthe constant matrix B

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 ______.
Answerdoes not exist

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.

Answeridentity

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.

AnswerFalse

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.

AnswerFalse

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.

AnswerTrue

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.

AnswerFalse

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.

AnswerTrue

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
1Identity matrixAA= 0
2Singular matrixBA = Aᵀ
3Symmetric matrixCDiagonal entries = 1, rest = 0
4Skew-symmetric matrixDA≠ 0
5Non-singular matrixEAᵀ = −A
6Scalar matrixFAll diagonal entries equal, rest = 0
Answer

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 hasA≠ 0)

6 → F (Scalar matrix has equal diagonal entries and zeros elsewhere)

V. Answer Briefly

1. Define a matrix. What are the different types of matrices?
Answer

A matrix is a rectangular arrangement of numbers (or elements) into rows and columns, enclosed within brackets. It is denoted by a capital letter.

The different types of matrices are:

  • Row matrix: Has only one row (1 × n)
  • Column matrix: Has only one column (m × 1)
  • Square matrix: Number of rows equals number of columns (n × n)
  • Diagonal matrix: Square matrix with non-zero entries only on the main diagonal
  • Scalar matrix: Diagonal matrix with all diagonal entries equal
  • Identity (Unit) matrix: Diagonal matrix with all diagonal entries equal to 1
  • Zero (Null) matrix: All entries are zero
  • Symmetric matrix: A = Aᵀ
  • Skew-symmetric matrix: A = −Aᵀ
2. State the conditions under which two matrices can be multiplied.
Answer

Two matrices A and B can be multiplied to form the product AB only when the number of columns in A equals the number of rows in B.

If A is of order m × n and B is of order n × p, then AB exists and is of order m × p.

Note: Even if AB exists, BA may not exist (if p ≠ m), and even if both exist, AB ≠ BA in general.

3. State any four properties of determinants.
Answer

2. Row/column swap sign: If any two rows (or columns) are interchanged, the sign of the determinant changes. 3. Identical rows: If any two rows (or columns) are identical, the determinant is zero. 4. Scalar multiplication of a row: If every element of a row (or column) is multiplied by a scalar k, then the determinant is multiplied by k.

6. Triangular matrix: The determinant of a triangular matrix is the product of its diagonal elements.

1. **Aᵀ=A:** The determinant remains unchanged when rows and columns are interchanged (transpose).
5. Product rule:AB=A×B, provided A and B are square matrices of the same order.

(Any four of the above earn full marks.)

4. What is Cramer's rule? When is it applicable?
Answer

Cramer's rule is a method for solving a system of linear equations using determinants.

For the system AX = B, where A is the coefficient matrix:

xᵢ = |Aᵢ| / |A|
Here,Aᵢis the determinant obtained by replacing the i-th column of A with the constant column B.

Conditions for applicability:

  • The coefficient matrix A must be square (n equations, n unknowns).
  • The determinant |A| must be non-zero (|A| ≠ 0), meaning A is non-singular.
IfA= 0, Cramer's rule cannot be applied — the system either has no solution or infinitely many solutions.

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.

Worked Example:

Let A = [[2, 3], [1, 4]] and B = [[5, 0], [2, 1]]. Find AB and BA, and verify that AB ≠ BA.

Step 1: Find AB AB = [[2, 3], [[5, 0], [[2(5)+3(2), 2(0)+3(1)], [1, 4]]] × [2, 1]] = [1(5)+4(2), 1(0)+4(1)]] AB = [[10+6, 0+3], [[16, 3], [5+8, 0+4]] = [13, 4]] Step 2: Find BA BA = [[5, 0], [[2, 3], [[5(2)+0(1), 5(3)+0(4)], [2, 1]]] × [1, 4]] = [2(2)+1(1), 2(3)+1(4)]] BA = [[10+0, 15+0], [[10, 15], [4+1, 6+4]] = [ 5, 10]] Step 3: Compare AB and BA AB = [[16, 3] BA = [[10, 15] [13, 4]] [ 5, 10]] Since AB ≠ BA, matrix multiplication is NOT commutative.
**Verification of determinant propertyAB=A×B:**
|A| = (2)(4) − (3)(1) = 8 − 3 = 5 |B| = (5)(1) − (0)(2) = 5 − 0 = 5 |AB| = (16)(4) − (3)(13) = 64 − 39 = 25 |A| × |B| = 5 × 5 = 25 = |AB| ✓

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 determinantA.
2. IfA= 0, the inverse does not exist (A is singular).
3. IfA≠ 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.

Step 1: Compute |A| |A| = (3)(5) − (2)(7) = 15 − 14 = 1 Since |A| = 1 ≠ 0, the inverse exists. Step 2: Find adj(A) For A = [[a, b], [c, d]] adj(A) = [[d, −b], [−c, a]] adj(A) = [[5, −2], [−7, 3]] Step 3: Compute A⁻¹ A⁻¹ = (1/|A|) × adj(A) = (1/1) × [[5, −2], [−7, 3]] A⁻¹ = [[5, −2], [−7, 3]] Step 4: Verification — AA⁻¹ should equal I AA⁻¹ = [[3, 2], × [[5, −2], [[3(5)+2(−7), 3(−2)+2(3)], [7, 5]] [−7, 3]] = [7(5)+5(−7), 7(−2)+5(3)]] = [[15−14, −6+6], [35−35, −14+15]] = [[1, 0], [0, 1]] = I ✓

3. Solving a System of Linear Equations Using Cramer's Rule (with Worked Example)

Cramer's Rule Statement:

For the system:

a₁x + b₁y = c₁ a₂x + b₂y = c₂
x = |A₁|/|A|, y = |A₂|/|A| where |A| = a₁b₂ − a₂b₁ |A₁| = c₁b₂ − c₂b₁ |A₂| = a₁c₂ − a₂c₁

Worked Example:

Solve the system using Cramer's rule:

3x + 2y = 12 x + y = 5
Step 1: Write the coefficient matrix A and constant matrix B A = [[3, 2], [1, 1]] B = [[12], [ 5]] Step 2: Compute |A| |A| = (3)(1) − (1)(2) = 3 − 2 = 1 Since |A| = 1 ≠ 0, Cramer's rule is applicable and the system has a unique solution. Step 3: Compute |A₁| (replace column 1 with B) A₁ = [[12, 2], [ 5, 1]] |A₁| = (12)(1) − (5)(2) = 12 − 10 = 2 Step 4: Compute |A₂| (replace column 2 with B) A₂ = [[3, 12], [1, 5]] |A₂| = (3)(5) − (1)(12) = 15 − 12 = 3 Step 5: Find x and y x = |A₁|/|A| = 2/1 = 2 y = |A₂|/|A| = 3/1 = 3 Solution: x = 2, y = 3

Verification:

3(2) + 2(3) = 6 + 6 = 12 ✓ (satisfies equation 1) (2) + (3) = 5 ✓ (satisfies equation 2)

3×3 System Extension (Cramer's Rule):

For: a₁x + b₁y + c₁z = d₁ a₂x + b₂y + c₂z = d₂ a₃x + b₃y + c₃z = d₃ x = |A₁|/|A|, y = |A₂|/|A|, z = |A₃|/|A| where |A| is the determinant of the 3×3 coefficient matrix, |A₁| replaces column 1 of A with [d₁, d₂, d₃]ᵀ, |A₂| replaces column 2 of A with [d₁, d₂, d₃]ᵀ, |A₃| replaces column 3 of A with [d₁, d₂, d₃]ᵀ.

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)/ArequiresA≠ 0. Always computeAfirst and check it is non-zero before finding the inverse.
6. Wrong Cramer's rule substitution: InA₁, replace only column 1 with B, keeping all other columns exactly as in A. Similarly forA₂, replace only column 2. Do not shuffle multiple columns.
7. Scalar multiplication vs. determinant scaling: For an n × n matrix A,kA= kⁿA, not kA. The scalar multiplies every entry, so the factor appears n times (once per row) in the determinant.

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.

Frequently Asked Questions

For A = [[a,b],[c,d]], |A| = ad − bc. Multiply the main diagonal elements and subtract the product of the off-diagonal elements.