ICSE Class 10 Maths Matrices Solutions Step-by-Step
ICSE Class 10 Maths Matrices: what this chapter asks you to do
ICSE Class 10 Maths matrices questions test whether you can read the order of a matrix, compare corresponding elements, add or subtract conformable matrices, multiply matrices in the correct order and use the inverse of a 2 \times 2 matrix when the determinant is non-zero. This replacement page explains Selina Chapter 9 Matrices as a solution resource: first the rules, then worked examples, then exam-style checks and common errors.
The chapter is not about memorising many matrix forms. It is about applying a few rules without mixing up rows and columns. For syllabus alignment, check the CISCE official website. For related practice on this site, use ICSE Class 10 solutions, ICSE textbook solutions and ICSE question papers.
Concept snapshot: rows talk to columns
For matrix multiplication, imagine each row of the first matrix asking a question to each column of the second matrix. The answer is one entry in the product matrix. This is why AB can exist while BA may not exist, and why changing the order can change the answer.
Formula and method reference for Selina Chapter 9 Matrices
Use this table before you start any solution. Most mistakes in Concise Mathematics Selina Solutions Class 10 ICSE Chapter 9 Matrices come from applying a rule before checking the order.
| Idea | Rule | What to check first |
|---|---|---|
| Order of a matrix | A matrix with m rows and n columns has order m \times n. | Count rows first, columns second. |
| Equality | If A=B, both matrices must have the same order and corresponding elements must be equal. | Compare only matching positions. |
| Addition or subtraction | (a_{ij}) \pm (b_{ij})=(a_{ij}\pm b_{ij}). | The two matrices must have the same order. |
| Scalar multiplication | If k is a number, kA is formed by multiplying every element of A by k. | Apply k to every entry, including negative entries. |
| Matrix multiplication | If A is m \times n and B is n \times p, then AB is possible and has order m \times p. | Columns of A must equal rows of B. |
| Identity matrix | For a 2 \times 2 matrix, I=\begin{bmatrix}1&0\\0&1\end{bmatrix}, and AI=IA=A when multiplication is defined. | Use the identity matrix of matching order. |
| Inverse of 2 \times 2 matrix | If A=\begin{bmatrix}a&b\\c&d\end{bmatrix} and ad-bc \neq 0, then A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}. | The determinant ad-bc must not be zero. |
Examiner’s mindset: show the condition before the calculation
In board-style matrix answers, the working is usually credited in stages: checking whether an operation is possible, writing the correct element-wise or row-column step, and then giving the final matrix. The exact marking scheme can vary by paper, so do not assume a fixed mark count. Still, a clear condition such as “orders are different, so addition is not possible” is often the difference between a correct reason and an incomplete answer.
How to use Concise Mathematics Selina Solutions Class 10 ICSE Chapter 9 Matrices
Use the solutions below by question type. Printed editions may adjust numbering or options, but the methods for Selina Chapter 9 remain the same: equality, operations, multiplication, inverse and matrix equations.
- For equality questions: equate corresponding entries and solve the resulting linear equations.
- For addition and subtraction: first check that both matrices have the same order.
- For multiplication: check conformability, then multiply row by column.
- For inverse questions: calculate the determinant first; no inverse exists if the determinant is 0.
- For linear equations: form AX=B, then use X=A^{-1}B only when A^{-1} exists.
Exercise 9(A): matrix equality, types and basic operations
Exercise 9(A) in Selina-style matrices practice usually checks definitions, equality of matrices, transpose, addition, subtraction and additive inverse. These are direct questions, but they require exact comparison of positions.
Question 1: Solve the matrix equality
Problem: If \begin{bmatrix}x+2&7\\y+3&a-2\end{bmatrix}=\begin{bmatrix}4&b-3\\4&3\end{bmatrix}, find x, y, a and b.
Step 1: Since the matrices are equal, corresponding elements are equal.
x+2=4,\quad 7=b-3,\quad y+3=4,\quad a-2=3
Step 2: Solve each equation separately.
x=4-2=2
b=7+3=10
y=4-3=1
a=3+2=5
Final answer: x=2,\ y=1,\ a=5,\ b=10.
Question 2: Find an additive inverse
Problem: If A=\begin{bmatrix}4&2\\7&-2\end{bmatrix} and B=\begin{bmatrix}-2&1\\3&-4\end{bmatrix}, find the additive inverse of A+B.
Step 1: Add A and B element-wise because both are of order 2 \times 2.
A+B=\begin{bmatrix}4+(-2)&2+1\\7+3&-2+(-4)\end{bmatrix}=\begin{bmatrix}2&3\\10&-6\end{bmatrix}
Step 2: The additive inverse of a matrix M is -M.
-(A+B)=-\begin{bmatrix}2&3\\10&-6\end{bmatrix}=\begin{bmatrix}-2&-3\\-10&6\end{bmatrix}
Final answer: The additive inverse of A+B is \begin{bmatrix}-2&-3\\-10&6\end{bmatrix}.
Question 3: True or false with reasons
Statement: If A and B have orders 3 \times 2 and 2 \times 3, then A+B is possible.
Answer: False. Addition is possible only when the two matrices have the same order.
Statement: The transpose of a 2 \times 1 matrix is a 2 \times 1 matrix.
Answer: False. A 2 \times 1 matrix becomes a 1 \times 2 matrix after transposition.
Statement: The transpose of a square matrix is a square matrix.
Answer: True. If the order is n \times n, the transpose also has order n \times n.
Question 4: Find the transpose and related matrices
Problem: Let M=\begin{bmatrix}5&-3\\-2&4\end{bmatrix}. Find M^t, M+M^t and M^t-M.
Step 1: Interchange rows and columns to get the transpose.
M^t=\begin{bmatrix}5&-2\\-3&4\end{bmatrix}
Step 2: Add corresponding elements.
M+M^t=\begin{bmatrix}5&-3\\-2&4\end{bmatrix}+\begin{bmatrix}5&-2\\-3&4\end{bmatrix}=\begin{bmatrix}10&-5\\-5&8\end{bmatrix}
Step 3: Subtract M from M^t.
M^t-M=\begin{bmatrix}5&-2\\-3&4\end{bmatrix}-\begin{bmatrix}5&-3\\-2&4\end{bmatrix}=\begin{bmatrix}0&1\\-1&0\end{bmatrix}
Final answer: M^t=\begin{bmatrix}5&-2\\-3&4\end{bmatrix}, M+M^t=\begin{bmatrix}10&-5\\-5&8\end{bmatrix}, M^t-M=\begin{bmatrix}0&1\\-1&0\end{bmatrix}.
Exercise 9(B): matrix multiplication and conformability
Matrix multiplication is the part of Maths where students most often lose steps. Before multiplying, write the order of both matrices. If A is m \times n and B is n \times p, then AB is possible and the product is m \times p.
Question 5: Multiply two matrices
Problem: If A=\begin{bmatrix}2&-1&3\\0&4&5\end{bmatrix} and B=\begin{bmatrix}1&2\\-3&0\\4&-1\end{bmatrix}, find AB.
Step 1: Check the order. A is 2 \times 3 and B is 3 \times 2. Since the middle numbers match, AB is possible and has order 2 \times 2.
Step 2: Multiply each row of A by each column of B.
AB=\begin{bmatrix}2(1)+(-1)(-3)+3(4)&2(2)+(-1)(0)+3(-1)\\0(1)+4(-3)+5(4)&0(2)+4(0)+5(-1)\end{bmatrix}
AB=\begin{bmatrix}2+3+12&4+0-3\\0-12+20&0+0-5\end{bmatrix}=\begin{bmatrix}17&1\\8&-5\end{bmatrix}
Final answer: AB=\begin{bmatrix}17&1\\8&-5\end{bmatrix}.
Question 6: Show that multiplication is not always commutative
Problem: Take A=\begin{bmatrix}1&2\\0&1\end{bmatrix} and B=\begin{bmatrix}3&1\\2&0\end{bmatrix}. Compare AB and BA.
AB=\begin{bmatrix}1(3)+2(2)&1(1)+2(0)\\0(3)+1(2)&0(1)+1(0)\end{bmatrix}=\begin{bmatrix}7&1\\2&0\end{bmatrix}
BA=\begin{bmatrix}3(1)+1(0)&3(2)+1(1)\\2(1)+0(0)&2(2)+0(1)\end{bmatrix}=\begin{bmatrix}3&7\\2&4\end{bmatrix}
Final answer: AB \neq BA. Matrix multiplication is not commutative in general.
Exercise 9(C): inverse matrix and linear equations
In the inverse method, the first line should usually calculate the determinant. The inverse of A=\begin{bmatrix}a&b\\c&d\end{bmatrix} exists only when ad-bc\neq 0. This condition is an important edge case in ICSE Class 10 Maths matrices.
Question 7: Find the inverse of a 2 \times 2 matrix
Problem: Find the inverse of A=\begin{bmatrix}2&1\\5&3\end{bmatrix}.
Step 1: Find the determinant.
\det(A)=ad-bc=(2)(3)-(1)(5)=6-5=1
Step 2: Since \det(A)=1\neq 0, the inverse exists.
Step 3: Apply the inverse formula.
A^{-1}=\frac{1}{1}\begin{bmatrix}3&-1\\-5&2\end{bmatrix}
Final answer: A^{-1}=\begin{bmatrix}3&-1\\-5&2\end{bmatrix}.
Question 8: Solve simultaneous equations by the matrix method
Problem: Solve 2x+y=7 and x-y=2 by using matrices.
Step 1: Write the equations in matrix form AX=B.
\begin{bmatrix}2&1\\1&-1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}7\\2\end{bmatrix}
Step 2: Find the determinant of A=\begin{bmatrix}2&1\\1&-1\end{bmatrix}.
\det(A)=(2)(-1)-(1)(1)=-2-1=-3
Step 3: Since \det(A)\neq 0, find A^{-1}.
A^{-1}=\frac{1}{-3}\begin{bmatrix}-1&-1\\-1&2\end{bmatrix}=\begin{bmatrix}\frac{1}{3}&\frac{1}{3}\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}
Step 4: Multiply A^{-1} by B.
\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}\frac{1}{3}&\frac{1}{3}\\\frac{1}{3}&-\frac{2}{3}\end{bmatrix}\begin{bmatrix}7\\2\end{bmatrix}
\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}\frac{7}{3}+\frac{2}{3}\\\frac{7}{3}-\frac{4}{3}\end{bmatrix}=\begin{bmatrix}3\\1\end{bmatrix}
Final answer: x=3 and y=1.
Test Yourself and case-study based questions
Test Yourself questions usually mix two or more rules from the chapter. A case-study based question may present data in rows and columns and ask you to form a matrix, identify its order, or use matrix operations to compare entries. Read the labels of rows and columns before calculating.
Case-style model: A shop records the number of notebooks and pens sold in two days by the matrix S=\begin{bmatrix}12&18\\15&20\end{bmatrix}, where rows represent Day 1 and Day 2, and columns represent notebooks and pens. If the price matrix is P=\begin{bmatrix}30\\10\end{bmatrix}, find the sales amount for each day.
Step 1: Check the order. S is 2 \times 2 and P is 2 \times 1, so SP is possible and has order 2 \times 1.
Step 2: Multiply rows of S by the price column.
SP=\begin{bmatrix}12&18\\15&20\end{bmatrix}\begin{bmatrix}30\\10\end{bmatrix}=\begin{bmatrix}12(30)+18(10)\\15(30)+20(10)\end{bmatrix}
SP=\begin{bmatrix}360+180\\450+200\end{bmatrix}=\begin{bmatrix}540\\650\end{bmatrix}
Final answer: Sales amount is ₹540 on Day 1 and ₹650 on Day 2.
Quick answer index for revision
This answer index lists the worked models on this page. Use it only after you have checked the steps above.
| Section | Question type | Final answer |
|---|---|---|
| Exercise 9(A) | Matrix equality | x=2,\ y=1,\ a=5,\ b=10 |
| Exercise 9(A) | Additive inverse | \begin{bmatrix}-2&-3\\-10&6\end{bmatrix} |
| Exercise 9(A) | Transpose | M^t=\begin{bmatrix}5&-2\\-3&4\end{bmatrix} |
| Exercise 9(B) | Matrix multiplication | AB=\begin{bmatrix}17&1\\8&-5\end{bmatrix} |
| Exercise 9(C) | Inverse matrix | A^{-1}=\begin{bmatrix}3&-1\\-5&2\end{bmatrix} |
| Exercise 9(C) | Linear equations by matrices | x=3,\ y=1 |
| Case-study model | Data matrix and price matrix | \begin{bmatrix}540\\650\end{bmatrix} |
Common mistakes students make in matrices
- Writing the order backwards: The order is rows first and columns second. A matrix with 2 rows and 3 columns is 2 \times 3, not 3 \times 2.
- Adding matrices of different orders: Addition and subtraction need the same order. If one matrix is 3 \times 2 and the other is 2 \times 3, the operation is not possible.
- Multiplying corresponding entries instead of row by column: For AB, each entry is made from a row of A and a column of B, not by simply multiplying entries in the same positions.
- Forgetting the determinant condition: A 2 \times 2 matrix has an inverse only if ad-bc\neq 0.
- Changing the order of multiplication: In general, AB\neq BA. Do not swap the matrices unless a question proves that both products are equal.
Frequently Asked Questions
What is conformability in ICSE Class 10 Maths matrices?
Conformability means that the orders of matrices allow a particular operation. For addition or subtraction, the orders must be the same. For multiplication AB, the number of columns of A must equal the number of rows of B.
How do I check matrix multiplication in Selina Chapter 9?
Write the order of both matrices first. If A is m \times n and B is n \times p, then AB is possible and its order is m \times p. Then multiply each row of A by each column of B.
When does the inverse of a 2 \times 2 matrix not exist?
The inverse does not exist when the determinant is zero. For A=\begin{bmatrix}a&b\\c&d\end{bmatrix}, calculate ad-bc. If ad-bc=0, A^{-1} is not defined.
Are matrix addition and matrix multiplication both commutative?
Matrix addition is commutative when both matrices have the same order, so A+B=B+A. Matrix multiplication is not commutative in general; AB and BA may be different or one of them may not even be defined.
How should I write answers for Concise Mathematics Selina Solutions Class 10 ICSE Chapter 9 Matrices?
Write the condition first, then show the calculation. For example, before multiplying, mention the orders and say whether the product is possible. For inverse questions, calculate ad-bc before writing the inverse formula.