Mathematics Class XII: Matrices | Concept & Types

Mathematics Class XII: Matrices
Concept, Notation, Order, and Types of Matrices

Topic-wise Comprehensive Notes

Concept 1: Definition, Notation, and Order of Matrices

📘 Definition

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

1.1 Notation

Matrices are usually denoted by capital letters ($A, B, C, \dots$). A general matrix $A$ of size $m \times n$ is represented as: \[ A = [a_{ij}]_{m \times n} \] where $i$ denotes the row ($1 \le i \le m$) and $j$ denotes the column ($1 \le j \le n$).

1.2 Order of a Matrix

A matrix having $m$ rows and $n$ columns is called a matrix of order $m \times n$ (read as $m$ by $n$).

Total number of elements in an $m \times n$ matrix is $mn$.
The element $a_{ij}$ is found at the intersection of the $i^{th}$ row and $j^{th}$ column.

1.3 Equality of Matrices

Two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ are said to be equal ($A = B$) if:

They are of the same order.
Each element of $A$ is equal to the corresponding element of $B$ (i.e., $a_{ij} = b_{ij}$ for all $i$ and $j$).

Solved Examples: Concept 1

📌 Example 1:

Construct a $2 \times 2$ matrix $A = [a_{ij}]$ whose elements are given by $a_{ij} = \frac{(i + 2j)^2}{2}$.

Solution: For $2 \times 2$ matrix, $i \in \{1,2\}$, $j \in \{1,2\}$.
$a_{11} = \frac{(1+2)^2}{2} = \frac{9}{2}$, $a_{12} = \frac{(1+4)^2}{2} = \frac{25}{2}$,
$a_{21} = \frac{(2+2)^2}{2} = \frac{16}{2}=8$, $a_{22} = \frac{(2+4)^2}{2} = \frac{36}{2}=18$.
Thus, $A = \begin{bmatrix} 9/2 & 25/2 \\ 8 & 18 \end{bmatrix}$.

📌 Example 2:

If a matrix has 18 elements, what are the possible orders it can have? What if it has 5 elements?

Solution: Possible orders: factor pairs of 18 → $(1\times18), (2\times9), (3\times6), (6\times3), (9\times2), (18\times1)$.
For 5 elements (prime): possible orders $(1\times5)$ and $(5\times1)$.

📌 Example 3:

Find $x, y, z$ and $a$ if: $\begin{bmatrix} x+3 & 2y+x \\ z-1 & 4a-6 \end{bmatrix} = \begin{bmatrix} 0 & -7 \\ 3 & 2a \end{bmatrix}$.

Solution: Equating entries: $x+3=0 \Rightarrow x=-3$; $2y+x = -7 \Rightarrow 2y-3=-7 \Rightarrow y=-2$; $z-1=3 \Rightarrow z=4$; $4a-6=2a \Rightarrow 2a=6 \Rightarrow a=3$. So $x=-3, y=-2, z=4, a=3$.

📌 Example 4:

How many matrices of order $2 \times 2$ can be formed using entries $\{0,1,2\}$?

Solution: $2\times2$ matrix has $4$ entries, each with $3$ choices → $3^4 = 81$ matrices.

📌 Example 5:

If $\begin{bmatrix} x-y & 2x+z \\ 2x-y & 3z+w \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$, find $x+y+z+w$.

Solution: From $(2,1): 2x-y=0 \Rightarrow y=2x$. $(1,1): x-y=-1 \Rightarrow x-2x=-1 \Rightarrow x=1$, so $y=2$. $(1,2): 2x+z=5 \Rightarrow 2+z=5 \Rightarrow z=3$. $(2,2): 3z+w=13 \Rightarrow 9+w=13 \Rightarrow w=4$. Sum $=1+2+3+4=10$.

Concept 2: Types of Matrices

Row Matrix: A matrix having only one row. Order: $1 \times n$.
Column Matrix: A matrix having only one column. Order: $m \times 1$.
Square Matrix: Number of rows equals number of columns ($m=n$).
Diagonal Matrix: Square matrix where $a_{ij}=0$ for $i \ne j$.
Scalar Matrix: Diagonal matrix with all diagonal entries equal: $a_{ii}=k$.
Identity Matrix ($I$): Scalar matrix with $k=1$.
Zero Matrix ($O$): All entries are zero.

Representation of a Diagonal Matrix (non-zero entries on main diagonal)

Solved Examples: Concept 2

📌 Example 1:

Is every scalar matrix an identity matrix? Explain.

Solution: No. A scalar matrix is diagonal with $a_{ii}=k$ (any constant). Identity matrix requires $k=1$. Eg. $\text{diag}(5,5,5)$ is scalar but not identity.

📌 Example 2:

Given $A$ is square with $a_{ij}=0$ when $i\ne j$ and $a_{ij}=i^2$ when $i=j$ for order $3\times3$. Identify type.

Solution: Diagonal matrix because off-diagonals zero. $A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 9 \end{bmatrix}$.

📌 Example 3:

For what $x,y$ is $A$ zero matrix? $A = \begin{bmatrix} x^2-1 & y+2 \\ 0 & x-1 \end{bmatrix}$.

Solution: $x^2-1=0\Rightarrow x=\pm1$, $y+2=0\Rightarrow y=-2$, $x-1=0\Rightarrow x=1$. So $x=1, y=-2$.

📌 Example 4:

Define a Square Matrix of order $n$. How many elements?

Solution: $n\times n$ matrix. Total elements = $n^2$.

📌 Example 5:

Construct $3\times1$ column matrix $A=[a_{i1}]$ where $a_{i1}=(-1)^i + i$ for $i=1,2,3$. Find sum of elements and express as sum of two matrices.

Solution: $a_{11}=(-1)^1+1=0$, $a_{21}=(-1)^2+2=3$, $a_{31}=(-1)^3+3=2$. Thus $A=\begin{bmatrix}0\\3\\2\end{bmatrix}$. Sum = $5$.
$A = \begin{bmatrix}(-1)^1\\(-1)^2\\(-1)^3\end{bmatrix} + \begin{bmatrix}1\\2\\3\end{bmatrix} = \begin{bmatrix}-1\\1\\-1\end{bmatrix} + \begin{bmatrix}1\\2\\3\end{bmatrix}$.

Quick reference — Matrix notation & equality

\[ A_{m\times n} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} \] \[ \text{If } A = B \;\text{ then }\; a_{ij}=b_{ij} \;\forall i,j \quad\text{and same order.} \]

Self Exercise

Find the number of all possible matrices of order 3 × 3 with each entry 0 or 1 such that the matrix is a diagonal matrix.
If A = [a_ij] is a square matrix of order 3 such that a_ij = i² – j², identify whether A is a null matrix, diagonal matrix, or neither. Justify.
If a matrix has p elements where p is a prime number, what are the possible orders?
Construct a 3 × 2 matrix A such that a_ij = |i – 3j| / 2.
Given
a-b 2a+c 2a-b 3c+d
=
-1 5 0 2
, find a, b, c, d.
Under what condition on i and j is the element a_ij of a square matrix A called a diagonal element?
A matrix A has x rows and x+5 columns. A matrix B has y rows and 11-y columns. If both matrices are equal, find x and y.
If A = [a_ij] is a scalar matrix of order n × n such that a_ii = k, find the sum of all elements of the matrix.
Can a 2 × 3 matrix be equal to a 3 × 2 matrix? Why or why not?
Provide an example of a matrix which is both a row matrix and a column matrix.

Solutions: Self Exercise

Solution 1: In a diagonal matrix, all non-diagonal entries are 0. Only the 3 diagonal entries (a₁₁, a₂₂, a₃₃) can be 0 or 1. Since each of these 3 positions has 2 choices, the total number of matrices = 2 × 2 × 2 = 8.
Solution 2: For a diagonal matrix, a_ij must be 0 for i ≠ j. Here, if i=1, j=2, a₁₂ = 1² – 2² = -3 (not zero). Since non-diagonal elements are not zero, it is neither. It is not a null matrix because a₁₂ ≠ 0.
Solution 3: A matrix with p elements where p is prime can only have factors 1 and p. Therefore, the possible orders are 1 × p or p × 1.
Solution 4: Calculating a_ij = |i – 3j| / 2 for a 3 × 2 matrix:
a₁₁=|1-3|/2 = 1; a₁₂=|1-6|/2 = 2.5
a₂₁=|2-3|/2 = 0.5; a₂₂=|2-6|/2 = 2
a₃₁=|3-3|/2 = 0; a₃₂=|3-6|/2 = 1.5
A =
1.0 2.5 0.5 2.0 0.0 1.5
Solution 5: By equating elements:
1) a – b = -1
2) 2a – b = 0 ⇒ b = 2a
Substitute b in (1): a – 2a = -1 ⇒ -a = -1 ⇒ a = 1. Then b = 2.
3) 2(1) + c = 5 ⇒ c = 3.
4) 3(3) + d = 2 ⇒ 9 + d = 2 ⇒ d = -7.
Solution 6: An element a_ij is called a diagonal element if i = j.
Solution 7: For two matrices to be equal, they must have the same order. Matrix A has order x × (x + 5) and matrix B has order y × (11 − y). Therefore, the number of rows and columns must be equal: x = y and x + 5 = 11 − y ⇒ Substituting x = y in second equation: x + 5 = 11 − x ⇒ 2x = 6 ⇒ x = 3 Since x = y ⇒ y = 3 Final Answer: x = 3, y = 3
Solution 8: In a scalar matrix, all n diagonal elements are k and all other elements are 0. Sum = n × k = nk.
Solution 9: No. Two matrices can only be equal if they have the same order. A 2 × 3 matrix has a different shape/dimension than a 3 × 2 matrix.
Solution 10: A 1 × 1 matrix, such as [5], is both a row matrix (one row) and a column matrix (one column).

Multiple Choice Questions: Matrices

A matrix in which the number of rows is equal to the number of columns is called:
1. Row matrix
2. Column matrix
3. Square matrix
4. Diagonal matrix
The number of elements in a matrix of order 3 × 4 is:
1. 7
2. 12
3. 1
4. None of these
For a square matrix A = [a_ij] to be a diagonal matrix, which condition must be satisfied?
1. a_ij = 0 for i = j
2. a_ij = 0 for i ≠ j
3. a_ij = 1 for i ≠ j
4. a_ij ≠ 0 for i ≠ j
A diagonal matrix in which all diagonal elements are equal is called a:
1. Identity matrix
2. Unit matrix
3. Scalar matrix
4. Null matrix
If a matrix has 7 elements, how many possible orders can it have?
1. 1
2. 2
3. 3
4. 4
The identity matrix of order 2 is:
1. [0 1; 1 0]
2. [1 0; 0 1]
3. [1 1; 1 1]
4. [0 0; 0 0]
In a 3 × 3 matrix, the total number of diagonal elements is:
1. 9
2. 6
3. 3
4. 1
If A is a 2 × 3 matrix and B is a 2 × 3 matrix, then A and B are:
1. Equal matrices
2. Comparable matrices
3. Square matrices
4. Identity matrices
The element a₂₃ in a matrix represents:
1. 2nd Column, 3rd Row
2. 3rd Row, 2nd Column
3. 2nd Row, 3rd Column
4. None of these
A null matrix of order 2 × 2 is also known as:
1. Zero matrix
2. Identity matrix
3. Scalar matrix
4. Diagonal matrix
What is the order of a matrix that has only one row and multiple columns?
1. m × 1
2. 1 × n
3. n × n
4. 1 × 1
Which of the following is a 1 × 1 matrix?
1. [0 0]
2. [1; 1]
3. [5]
4. None of these
Two matrices A and B are equal if and only if:
1. They have same order
2. Corresponding elements are equal
3. Both a and b
4. They are square matrices
If A = [a_ij] is a 3 × 3 matrix where a_ij = 0 for all i, j, it is a:
1. Unit matrix
2. Scalar matrix
3. Null matrix
4. Diagonal matrix
If a square matrix has 25 elements, its order is:
1. 5 × 5
2. 25 × 1
3. 1 × 25
4. 5 × 1

Answer Key: Matrices MCQ

1. (c) Square matrix
2. (b) 12
3. (b) a_ij = 0 for i ≠ j
4. (c) Scalar matrix
5. (b) 2
6. (b) [1 0; 0 1]
7. (c) 3
8. (b) Comparable
9. (c) 2nd Row, 3rd Col
10. (a) Zero matrix
11. (b) 1 × n
12. (c) [5]
13. (c) Both a and b
14. (c) Null matrix
15. (a) 5 × 5

Mathematics Class XII: Matrices Concept, Notation, Order, and Types of Matrices