Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
How to Use This Page
This page is your complete study companion for Chapter 1 — Applications of Matrices and Determinants. Start with the Key Concepts table to refresh definitions, then work through the MCQs, fill-in-the-blanks, and true-or-false sections for quick practice. Use the match-the-following exercise to strengthen associations. The brief and detailed answer sections mirror the 2-mark and 5-mark questions you will face in the board exam. Finally, review the Common Mistakes and Exam Checklist before every test to avoid losing marks.
Chapter 1 — Exercise Solutions Guide
This chapter is scoring if the row-column order and signs are handled carefully. Most mistakes happen while finding cofactors, adjoint, and determinant values.
Key Concepts (Quick Recap)
| Term | Definition | ||||
|---|---|---|---|---|---|
| Matrix | A rectangular array of numbers arranged in rows and columns | ||||
| Order of a matrix | Number of rows multiplied by number of columns; written m x n | ||||
| Square matrix | A matrix where the number of rows equals the number of columns | ||||
| Determinant | A scalar value computed from a square matrix; written as \ | A\ | |||
| Minor | Determinant of the sub-matrix obtained by deleting a row and column | ||||
| Cofactor | Minor multiplied by (-1)^(i+j); accounts for sign alternation | ||||
| Adjoint | Transpose of the cofactor matrix; adj(A) = (C_ij)^T | ||||
| Inverse of a matrix | A^-1 = adj(A) / \ | A\ | ; exists only when \ | A\ | is not zero |
| Singular matrix | A square matrix whose determinant is zero; has no inverse | ||||
| Non-singular matrix | A square matrix whose determinant is non-zero; has an inverse | ||||
| Cramer's Rule | Method to solve AX = B using ratios of determinants | ||||
| Consistent system | A system of equations that has at least one solution | ||||
| Inconsistent system | A system of equations that has no solution | ||||
| Rank of a matrix | The maximum number of linearly independent rows (or columns) | ||||
| Homogeneous system | A system AX = 0 where the constant column is zero |
Formula Recap
| Concept | Result | ||
|---|---|---|---|
| Matrix product | If A is m x n and B is n x p, then AB is m x p | ||
| Inverse condition | A inverse exists only when \ | A\ | is not 0 |
| Inverse formula | A inverse = adj A / \ | A\ | |
| Cramer's rule | x = Dx / D, y = Dy / D, z = Dz / D |
I. Choose the Best Answer (MCQs)
| 1. If A is a 3 x 3 matrix and \ | A\ | = 5, then \ | 2A\ | is |
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2. If A is a non-singular matrix, then A^-1 equals
(d) -adj(A)
| (a) adj(A) / \ | A\ | |
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| (b) \ | A\ | / adj(A) |
| (c) adj(A) times \ | A\ | |
| **Answer: (a) adj(A) / \ | A\ | ** (This is the standard formula for the inverse of a non-singular matrix.) |
| 4. If \ | A\ | = 0, then A is called a |
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| 7. If A and B are square matrices of the same order, then \ | AB\ | equals | ||
|---|---|---|---|---|
| (a) \ | A\ | + \ | B\ | |
| (b) \ | A\ | - \ | B\ | |
| (c) \ | A\ | times \ | B\ | |
| (d) \ | A\ | / \ | B\ | |
| **Answer: (c) \ | A\ | times \ | B\ | ** (The determinant of a product equals the product of the determinants.) |
8. A system of three linear equations in three unknowns has a unique solution if
(c) The rank of A is less than 3 (d) The equations are dependent
| (a) \ | A\ | = 0 |
|---|---|---|
| (b) \ | A\ | is not equal to 0 |
| **Answer: (b) \ | A\ | is not equal to 0** (A non-zero determinant means the coefficient matrix is non-singular, giving a unique solution.) |
II. Fill in the Blanks
| 2. If \ | A\ | is not equal to zero, then the matrix A is called a ______ matrix. |
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| 4. A^-1 = adj(A) / \ | A\ | is valid only when \ | A\ | is ______ zero. |
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| 10. For a system AX = B to have infinitely many solutions, the rank of A must be ______ the rank of the augmented matrix [A | B]. |
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III. True or False
1. Swapping two rows of a determinant changes the sign of the determinant. True. (Each row interchange multiplies the determinant by -1.)
2. A singular matrix has an inverse. False. Correction: A singular matrix has determinant zero and therefore has no inverse.
3. The adjoint of a matrix is always equal to its inverse.
| False. Correction: adj(A) = \ | A\ | times A^-1; it equals the inverse only when \ | A\ | = 1. |
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True. (When the determinant is zero, we must check the rank to decide.)
| 4. If \ | A\ | = 0, the system AX = B may be inconsistent or have infinitely many solutions. |
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5. The determinant of a diagonal matrix is the product of its diagonal elements. True. (All off-diagonal terms are zero, so only the diagonal product remains.)
6. Cramer's rule can be applied when the determinant of the coefficient matrix is zero. False. Correction: Cramer's rule requires a non-zero determinant; division by zero is undefined.
7. The cofactor of an element and its minor always have the same value. False. Correction: The cofactor may differ in sign from the minor depending on the position (i+j).
8. A 2 x 3 matrix has a determinant. False. Correction: Only square matrices have determinants; a 2 x 3 matrix is not square.
IV. Match the Following
| Column A | Column B | ||||||
|---|---|---|---|---|---|---|---|
| 1. Singular matrix | (c) Determinant is zero | ||||||
| 2. Adjoint of A | (e) Transpose of cofactor matrix | ||||||
| 3. \ | AB\ | (a) \ | A\ | times \ | B\ | ||
| 4. Cramer's rule | (d) x = Delta_x / Delta | ||||||
| 5. Inverse exists when | (b) \ | A\ | is not zero | ||||
| 6. Homogeneous system | (f) AX = 0 | ||||||
| 7. Row operation that preserves determinant | (h) R_i -> R_i + k R_j | ||||||
| 8. \ | kA\ | for 3 x 3 matrix | (g) k^3 \ | A\ |
Solving Linear Equations by Matrix Method
1. Write the equations in the form AX = B.
4. Use X = A inverse B. 5. Write x, y, z clearly as the final answer.
| 2. Find determinant \ | A\ | . |
|---|---|---|
| 3. If \ | A\ | is not zero, find adj A. |
Worked Pattern
For equations:
Write
Then:
Since determinant is not zero, inverse exists. Find A inverse and multiply with B to get x and y.
Worked Examples
Example 1 — Solve by matrix inverse method (completed)
Solve 2x + y = 7, x − y = 2.
So x = 3, y = 1. Check: 2(3) + 1 = 7 ✓ and 3 − 1 = 2 ✓.
Example 2 — Cramer's Rule
Solve x + 2y = 4, 3x − y = 5.
So x = 2, y = 1.
V. Answer Briefly (2-Mark Questions)
1. Define the determinant of a 2 x 2 matrix and give an example.
The determinant of a 2 x 2 matrix [[a, b], [c, d]] is computed as ad - bc. For example, the determinant of [[3, 2], [1, 4]] is (3)(4) - (2)(1) = 12 - 2 = 10. It represents a scalar value associated with the matrix and determines whether the matrix is singular or non-singular.
2. State the condition for the inverse of a matrix to exist.
| A matrix A has an inverse if and only if its determinant \ | A\ | is not equal to zero. Such a matrix is called non-singular. The inverse is given by A^-1 = adj(A) / \ | A\ | . If the determinant is zero, the matrix is singular and no inverse exists. |
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3. What is the adjoint of a matrix? How is it related to the inverse?
| The adjoint of a matrix A is the transpose of its cofactor matrix. Each cofactor C_ij = (-1)^(i+j) times the minor M_ij. The adjoint is used in computing the inverse through the formula A^-1 = adj(A) / \ | A\ | . Without the adjoint, the inverse cannot be computed directly. |
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4. Write two properties of determinants.
(i) If two rows (or columns) of a determinant are interchanged, the sign of the determinant changes. (ii) If two rows (or columns) are identical, the value of the determinant is zero. These properties are useful for simplifying determinant calculations and checking for singular matrices.
5. What is Cramer's rule? When can it be applied?
Cramer's rule is a method to solve a system of linear equations using determinants. For a system AX = B, each variable is found by dividing a modified determinant by the main determinant: x = Delta_x / Delta, y = Delta_y / Delta, etc. It can be applied only when the determinant of the coefficient matrix is non-zero.
6. Distinguish between a consistent and an inconsistent system of equations.
A consistent system has at least one solution; the equations intersect at one point or overlap completely. An inconsistent system has no solution; the equations represent parallel lines that never meet. Checking the determinant and the rank of the augmented matrix helps determine consistency.
VI. Answer in Detail (5-Mark Questions)
1. Explain the matrix method for solving a system of linear equations with an example.
| To solve a system of linear equations using the matrix method, first express the system in the form AX = B, where A is the coefficient matrix, X is the column of variables, and B is the column of constants. Next, compute the determinant \ | A\ | . If \ | A\ | is not zero, the system has a unique solution found using X = A^-1 B. Calculate adj(A) by finding the cofactor matrix and transposing it. Then compute A^-1 = adj(A) / \ | A\ | . Multiply A^-1 with B to obtain the values of the variables. For example, to solve 2x + y = 7 and x - y = 2, form A = [[2, 1], [1, -1]] and B = [[7], [2]]. Since \ | A\ | = -3 which is non-zero, the inverse exists. Find A^-1 = (1/3)[[1, 1], [1, -2]] and multiply by B to get x = 3, y = 1. Always verify by substituting back into the original equations. |
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2. Describe Cramer's rule for solving a system of two equations in two unknowns and illustrate with an example.
| Cramer's rule provides a direct formula for solving linear systems using determinants. For two equations a_1 x + b_1 y = c_1 and a_2 x + b_2 y = c_2, define Delta = \ | A\ | = a_1 b_2 - a_2 b_1. Then Delta_x replaces the first column of A with the constants: Delta_x = c_1 b_2 - c_2 b_1. Similarly, Delta_y replaces the second column: Delta_y = a_1 c_2 - a_2 c_1. The solutions are x = Delta_x / Delta and y = Delta_y / Delta, provided Delta is not zero. As an illustration, solve x + 2y = 4 and 3x - y = 5. We get Delta = (1)(-1) - (3)(2) = -7, Delta_x = (4)(-1) - (5)(2) = -14, Delta_y = (1)(5) - (3)(4) = -7. Therefore x = -14 / -7 = 2 and y = -7 / -7 = 1. If Delta equals zero, the system is either inconsistent or has infinitely many solutions. |
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3. Explain how to find the inverse of a 3 x 3 matrix step by step.
| Finding the inverse of a 3 x 3 matrix involves several systematic steps. First, verify that the determinant is non-zero by expanding along any row or column. If \ | A\ | = 0, the inverse does not exist. Next, compute the minor of each element by deleting the corresponding row and column and finding the 2 x 2 determinant. Then find the cofactor of each element using C_ij = (-1)^(i+j) M_ij. Arrange all cofactors in a matrix to form the cofactor matrix. Transpose the cofactor matrix to get the adjoint, adj(A). Finally, divide each element of adj(A) by the determinant: A^-1 = adj(A) / \ | A\ | . For accuracy, verify that A times A^-1 equals the identity matrix. Common errors include forgetting to transpose the cofactor matrix, mixing up signs in cofactors, and making arithmetic mistakes in the 2 x 2 minor calculations. Systematic row-column tracking helps prevent these errors. |
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4. Discuss the consistency of a system of three linear equations in three unknowns.
| A system of three linear equations in three unknowns, written as AX = B, can be classified based on the determinant and ranks. If \ | A\ | is not zero, the system has a unique solution given by X = A^-1 B; it is consistent. If \ | A\ | = 0, we must compare the rank of the coefficient matrix A with the rank of the augmented matrix [A | B]. If rank(A) = rank([A | B]) = number of variables, the system is consistent with a unique solution. If rank(A) = rank([A | B]) is less than the number of variables, the system has infinitely many solutions and is still consistent. If rank(A) is not equal to rank([A | B]), the system is inconsistent and has no solution. For homogeneous systems (B = 0), the trivial solution x = y = z = 0 always exists. A non-trivial solution exists only when \ | A\ | = 0. This analysis helps determine not just whether solutions exist, but also how many. |
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5. State and prove six important properties of determinants.
| Property 1: The value of a determinant remains unchanged if its rows and columns are interchanged (i.e., \ | A\ | = \ | A^T\ | ). Property 2: If two rows (or columns) are interchanged, the sign of the determinant changes. Property 3: If two rows (or columns) are identical, the determinant is zero. Property 4: If each element of a row (or column) is multiplied by k, the determinant is multiplied by k. Property 5: If a multiple of one row is added to another row, the determinant value is unchanged. Property 6: If each element of a row can be expressed as the sum of two terms, the determinant can be expressed as the sum of two determinants. These properties are fundamental to simplifying determinant calculations. For example, using Property 5 we can create zeros in a row or column before expanding, which greatly reduces computation. Using Property 4, if we factor out common elements from a row, the arithmetic becomes simpler. Mastery of these properties is essential for efficiently solving problems involving determinants. |
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Important Question Types
- Find the inverse of a 2 x 2 or 3 x 3 matrix.
- Solve simultaneous equations using matrix method.
- Find determinant using expansion.
- Check whether a system is consistent.
- Use matrix method for application problems.
Common Mistakes to Avoid
1. ❌ Writing adj(A) as the cofactor matrix without transposing. ✅ adj(A) is the transpose of the cofactor matrix. Always transpose after computing cofactors.
2. ❌ Forgetting the (-1)^(i+j) sign in cofactor calculations. ✅ Use the sign pattern [[+, -, +], [-, +, -], [+, -, +]] for a 3 x 3 matrix to get every cofactor sign correct.
3. ❌ Interchanging rows and columns while forming the coefficient matrix A. ✅ Write the equations in standard form first (ax + by + cz = d), then extract coefficients row by row.
4. ❌ Not checking whether the determinant is zero before computing the inverse.
| ✅ Always compute \ | A\ | first. If it is zero, the inverse does not exist and the system may be inconsistent or have infinite solutions. |
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| 5. ❌ Multiplying only the first row by k when computing \ | kA\ | for an n x n matrix. | ||||||
|---|---|---|---|---|---|---|---|---|
| ✅ Each of the n rows gets multiplied by k, so \ | kA\ | = k^n \ | A\ | . For a 3 x 3 matrix, \ | kA\ | = k^3 \ | A\ | . |
6. ❌ Setting up Cramer's rule by replacing the wrong column. ✅ To find Delta_x, replace only the x-column (first column) with the constants column. To find Delta_y, replace only the y-column (second column).
7. ❌ Making arithmetic errors in 2 x 2 minor calculations inside a 3 x 3 determinant. ✅ Double-check each minor by clearly crossing out the relevant row and column before computing the remaining 2 x 2 determinant.
8. ❌ Confusing consistent and inconsistent systems.
| ✅ Compare rank(A) with rank([A | B]). Equal ranks means consistent; unequal ranks means inconsistent. |
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Exam Checklist
- Verify the matrix order matches the number of equations and unknowns before starting.
- Compute the determinant of A first; if zero, state that the inverse does not exist.
- Use the correct sign pattern for cofactors in a 3 x 3 matrix.
- Remember to transpose the cofactor matrix to obtain the adjoint.
- Write the formula A^-1 = adj(A) / \|A\| before substituting values.
- In Cramer's rule, replace the correct column with the constants column.
- For consistency questions, find rank(A) and rank([A|B]) and compare them.
- Box or clearly underline final answers for x, y, z.
- Verify answers by substituting back into at least one original equation.
- For homogeneous systems, remember the trivial solution always exists; look for non-trivial solutions only when \|A\| = 0.
- State each step clearly: forming AX = B, finding \|A\|, computing adj(A), then X = A^-1 B.
- Check for sign errors in the final multiplication of A^-1 with B.