Matrices Class 10 ICSE — Bansal Chapter 9
The matrices class 10 ICSE chapter in Bansal’s Mathematics builds the subject from first principles. It defines a matrix, states its order, and names the seven types you must recognise. The chapter then develops operations: transpose, equality, addition, subtraction, scalar and matrix multiplication. Each operation comes with a compatibility rule and a worked pattern, so you can tell when an operation is possible and carry it out correctly.
Download Matrices Class 10 ICSE Mathematics Chapter 9 PDF
Download the Matrices Class 10 ICSE Mathematics chapter PDF from the Bansal textbook to revise offline or print for practice. It is the full chapter as printed in the book — 20 pages, 14 sections, 63 questions.
What is inside Chapter 9
20 pages · 14 sections · 11 worked examples · 63 exercise questions
| Section | Title |
|---|---|
| 9.1 | Matrix |
| 9.2 | Order of a Matrix |
| 9.3 | Elements of a Matrix |
| 9.4 | Types of Matrices |
| 9.5 | Transpose of a Matrix |
| 9.6 | Equality of Matrices |
| 9.7 | Addition of Matrices |
| 9.8 | Subtraction of Matrices |
| 9.9 | Additive Identity |
| 9.10 | Additive Inverse |
| 9.11 | Solving Matrix Equations |
| 9.12 | Multiplication of a matrix by a scalar (real number) |
| 9.13 | Multiplication of Matrices |
| 9.14 | Identity Matrix for Multiplication |
- EXERCISE 9(B): 11 questions
- EXERCISE 9(C): 31 questions
- EXERCISE 9(D): 21 questions
9.1 Matrix
A matrix is a rectangular arrangement of numbers set out in rows and columns. The plural of matrix is matrices. Each number or entry inside the brackets is called an element. The horizontal lines are rows, while the vertical lines are columns.
For example, \begin{bmatrix} 4 & 7 & -2 \\ 3 & 0 & 5 \end{bmatrix} is a matrix with two rows and three columns. Reading it aloud helps: “row one is 4, 7, −2; row two is 3, 0, 5.” This vocabulary appears in every later section, so fix it before moving on.
Matrices appear throughout the Class 10 ICSE syllabus in various shapes. A matrix may have one row, one column, or many of both. The shape decides which operations you can perform later. You can find this chapter alongside its neighbours in the Class 10 ICSE Maths book collection.
9.2 Order of a Matrix
The order of a matrix is stated as rows first, then columns. If a matrix has m rows and n columns, its order is written m \times n and read “m by n.” The number of rows always comes before the number of columns.
Take a matrix with 2 rows and 4 columns. Its order is 2 \times 4. A matrix with 5 rows and 3 columns is written B_{5 \times 3} when named B. The little subscript is a compact way to record the order.
Why does order matter? Because addition and multiplication both check order before they allow an operation. If the orders do not match, the operation is impossible. Stating an order correctly is the first step of every matrix question in this ICSE chapter.
9.3 Elements of a Matrix
An element is each entry inside a matrix. The total number of elements equals the number of rows times the number of columns. A matrix with 3 rows and 5 columns holds 3 \times 5 = 15 elements.
One element count can produce several orders. A matrix with 12 elements could be 1 \times 12, 2 \times 6, 3 \times 4, 4 \times 3, 6 \times 2, or 12 \times 1. Each pair multiplies to 12, so all six are possible.
This matters when a question gives only the element count and asks for the order. You list the factor pairs of that count. Then you check any extra condition, such as “square matrix,” to pick the one valid order.
9.4 Types of Matrices
This section names the seven types you must recognise on sight. A row matrix has only one row and is also called a row vector. A column matrix has only one column and is also called a column vector. A square matrix has equal rows and columns. A rectangular matrix has unequal rows and columns.
A zero or null matrix has every element equal to zero. A diagonal matrix is square, with all elements zero except those on the leading diagonal. A unit or identity matrix is a diagonal matrix whose leading diagonal entries are all 1; it is denoted I. The leading diagonal runs from top left to bottom right.
The comparison table shows each type with an example and its order.
| Type | Example | Order |
|---|---|---|
| Row matrix | [katex display="true"]5 \quad 9 \quad 2[/katex] | 1 \times 3 |
| Column matrix | \begin{bmatrix} 4 \\ 7 \\ 1 \end{bmatrix} | 3 \times 1 |
| Square matrix | \begin{bmatrix} 2 & 5 \\ 3 & -1 \end{bmatrix} | 2 \times 2 |
| Rectangular matrix | \begin{bmatrix} 2 & 4 & 7 \\ 1 & 0 & 5 \end{bmatrix} | 2 \times 3 |
| Zero matrix | \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} | 2 \times 2 |
| Diagonal matrix | \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} | 2 \times 2 |
| Identity matrix | \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} | 2 \times 2 |
9.5 Transpose of a Matrix
The transpose of a matrix is formed by interchanging its rows and columns. If A is a matrix, its transpose is written A^t. Row one of A becomes column one of A^t, row two becomes column two, and so on.
Take A = \begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \end{bmatrix}, a 2 \times 3 matrix. Its transpose is A^t = \begin{bmatrix} 1 & 2 \\ 4 & 5 \\ 7 & 8 \end{bmatrix}, which has order 3 \times 2. The order flips: an m \times n matrix transposes to n \times m.
The transpose appears in several ICSE class 10 matrices questions. Some ask you to add a matrix to its own transpose. That is only possible when the order matches, which happens for square matrices.
9.6 Equality of Matrices
Two matrices are equal when both conditions hold. First, they must have the same order. Second, every corresponding element must be equal.
These conditions let you solve for unknown entries. Suppose \begin{bmatrix} 2a+1 & b \\ 3 & c-2 \end{bmatrix} = \begin{bmatrix} 7 & 4 \\ 3 & 1 \end{bmatrix}. Match each position: 2a+1 = 7, so a = 3. Also b = 4, and c-2 = 1, so c = 3.
The method is always the same: write one equation per matching position, then solve each one. Check the order first. If the two matrices differ in order, they can never be equal, no matter what the entries look like.
9.7 Addition of Matrices
Two matrices can be added only when they have the same order. This is the compatibility rule for addition. Adding means adding corresponding elements, and the result keeps that same order.
For example, \begin{bmatrix} 2 & 5 \\ 1 & 4 \end{bmatrix} + \begin{bmatrix} 3 & 1 \\ 6 & 2 \end{bmatrix} = \begin{bmatrix} 2+3 & 5+1 \\ 1+6 & 4+2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 7 & 6 \end{bmatrix}.
Add position by position: top-left with top-left, top-right with top-right, and so on. If the orders differ, the addition is impossible. The correct answer is “not possible” rather than a guessed matrix.
9.8 Subtraction of Matrices
Subtraction follows the same rule as addition. The two matrices must have the same order. Subtract corresponding elements, and the result keeps that order.
For example, \begin{bmatrix} 5 & 4 \\ 2 & 1 \end{bmatrix} - \begin{bmatrix} 3 & 0 \\ 4 & 2 \end{bmatrix} = \begin{bmatrix} 5-3 & 4-0 \\ 2-4 & 1-2 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ -2 & -1 \end{bmatrix}. Note that the bottom row gives negative results; that is normal.
The section also verifies two laws for addition. Addition is commutative, meaning A + B = B + A. It is also associative, meaning (A + B) + C = A + (B + C). These laws hold because adding is done entry by entry, and ordinary number addition obeys both.
9.9 Additive Identity
In ordinary numbers, adding zero leaves a number unchanged: 3 + 0 = 3. Zero is the additive identity for numbers.
The null matrix plays the same role for matrices. If you add a null matrix of the same order to any matrix, the matrix stays unchanged. For instance, \begin{bmatrix} 4 & 6 \\ 0 & 7 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 4 & 6 \\ 0 & 7 \end{bmatrix}.
So the null matrix of matching order is called the additive identity in matrices. Every entry of the null matrix is zero, which is why adding it changes nothing.
9.10 Additive Inverse
The additive inverse of a matrix A is the matrix -A, formed by negating every element of A. When you add A and -A, you get the null matrix of the same order.
Take A = \begin{bmatrix} 4 & -2 \\ -3 & 7 \end{bmatrix}. Its additive inverse is -A = \begin{bmatrix} -4 & 2 \\ 3 & -7 \end{bmatrix}. Adding them gives \begin{bmatrix} 4-4 & -2+2 \\ -3+3 & 7-7 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}.
This mirrors the number line, where 5 and −5 add to zero. The inverse reverses every sign, so each pair of corresponding elements cancels to zero.
9.11 Solving Matrix Equations
When a matrix equation has an unknown matrix X, you isolate X the same way you isolate a variable in algebra. If A + X = B, then X = B - A. If X - B = A, then X = A + B.
Work a 2 \times 2 example. Let A = \begin{bmatrix} 2 & 5 \\ -1 & 3 \end{bmatrix} and B = \begin{bmatrix} 7 & 1 \\ 4 & 0 \end{bmatrix}. For A + X = B, subtract: X = \begin{bmatrix} 7-2 & 1-5 \\ 4-(-1) & 0-3 \end{bmatrix} = \begin{bmatrix} 5 & -4 \\ 5 & -3 \end{bmatrix}.
Subtract entry by entry, and keep the order. The unknown matrix has the same order as A and B because only matrices of the same order can be added or subtracted.
9.12 Multiplication of a matrix by a scalar (real number)
Scalar multiplication means multiplying every element of the matrix by the same number. The order of the matrix does not change.
For example, 3\begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ 0 & 12 \end{bmatrix}. A fractional scalar works the same way: \frac{1}{2}\begin{bmatrix} 6 & -4 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 3 & -2 \\ 1 & 0 \end{bmatrix}.
Scalar multiplication also helps solve equations with a scalar factor. In A + 2X = B, first find B - A, then halve each entry to get X. The scalar 2 multiplies every element of X, so you undo it by dividing every element by 2.
9.13 Multiplication of Matrices
Matrix multiplication uses a compatibility rule. The product AB exists only when the number of columns in A equals the number of rows in B. Each entry of the product is a sum of products: row of A times column of B, added term by term.
Work a 2 \times 2 example. Let A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} and B = \begin{bmatrix} 2 & 0 \\ 1 & 5 \end{bmatrix}. Then:
AB = \begin{bmatrix} 1(2)+2(1) & 1(0)+2(5) \\ 3(2)+4(1) & 3(0)+4(5) \end{bmatrix} = \begin{bmatrix} 4 & 10 \\ 10 & 20 \end{bmatrix}
The section draws three key conclusions. Matrix multiplication is not commutative: AB \neq BA in general. The product of two nonzero matrices can be the zero matrix. And the cancellation law fails: AB = AC does not force B = C.
Common mistakes and corrections:
| Mistake | Correction |
|---|---|
| Multiplying any two matrices | Check that columns of A equal rows of B first |
| Assuming AB equals BA | Compute BA separately; the results usually differ |
| Cancelling A from AB = AC | Cancellation does not apply to matrix multiplication |
| Writing (A+B)(A-B) = A^2 - B^2 | Expand to AB and BA; they differ, so the identity fails |
For the other chapters that build on these skills, browse the Class 10 ICSE books list.
9.14 Identity Matrix for Multiplication
The unit matrix I is the identity for multiplication. For any matrix A of the same order as I, A \times I = A = I \times A. The identity matrix is a diagonal matrix with 1s on the leading diagonal.
The section also gives the general order rule. If A is m \times n and B is n \times p, then the product AB is possible and has order m \times p. Write it as A_{m \times n} \times B_{n \times p} = C_{m \times p}. The inner numbers n must match; the outer numbers m and p give the product’s order.
You can also evaluate expressions involving I. Take A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}. First A^2 = \begin{bmatrix} 1 & 8 \\ 0 & 9 \end{bmatrix}.
Then A^2 - 3A + 2I = \begin{bmatrix} 1 & 8 \\ 0 & 9 \end{bmatrix} - \begin{bmatrix} 3 & 6 \\ 0 & 9 \end{bmatrix} + \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 2 \\ 0 & 2 \end{bmatrix}.
What to practise from Matrices Class 10 ICSE
The three exercise sets in the matrices class 10 ICSE chapter drill the operations above. Several questions carry board years spanning 2003 to 2015, marking past ICSE exam questions. Questions on matrix multiplication and solving for an unknown matrix recur across years.
- Exercise 9(B) — 11 questions on scalar multiplication and matrix equations
- Exercise 9(C) — 31 questions on matrix multiplication, powers, and expressions
- Exercise 9(D) — 21 questions mixing all operations with board-year questions
The official CISCE board website publishes the syllabus and exam format for Class 10 mathematics.
Matrices Class 10 ICSE frequently asked questions
When can two matrices be added or subtracted in matrices class 10 ICSE maths?
Only when they have the same order. Add or subtract corresponding elements, and the result keeps that order. If the orders differ, the operation is not possible.
Why is AB not equal to BA in Class 10 ICSE matrix multiplication?
Because the row-by-column process is not symmetric. In AB, you use rows of A with columns of B; in BA, you use rows of B with columns of A. These usually give different matrices, so multiplication is not commutative.
What is the difference between a diagonal matrix and an identity matrix in Class 10 ICSE Maths?
A diagonal matrix is square with nonzero entries only on the leading diagonal. An identity matrix is a special diagonal matrix where every leading-diagonal entry is 1. It is denoted I and leaves any matrix unchanged under multiplication.
How do I find the order of the unknown matrix X in a Class 10 ICSE product equation?
Use the order rule A_{m \times n} \times B_{n \times p} = C_{m \times p}. The inner number n must match. The outer numbers m and p give the product’s order, which tells you X’s order.
Does (A + B)(A – B) equal A squared minus B squared for matrices in Class 10 ICSE?
No. In ordinary algebra, (A+B)(A-B) = A^2 - AB + BA - B^2. Since AB and BA differ for matrices, the middle terms do not cancel. The identity fails for matrices.
Reference: ICSE Class 10 Mathematics textbook material, chapter 9: Matrices — CISCE-aligned.
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