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Q2:
If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
Solution :
Step 1: Theoretical Foundation of Matrix Order
An $m \times n$ matrix is defined by $m$ rows and $n$ columns. The total number of elements in such a matrix is given by the product of the number of rows and the number of columns, denoted as $N = m \times n$ [Per the definition of the order of a matrix]. To find all possible orders for a matrix with $N$ elements, we must identify all unique factor pairs $(m, n)$ of the integer $N$, where $m, n \in \mathbb{N}$.
Step 2: Analysis for 24 Elements
To find the possible orders for $N = 24$, we determine all positive integer factor pairs of 24. We systematically test divisors of 24:
- $24 \div 1 = 24 \implies (1 \times 24)$ and $(24 \times 1)$
- $24 \div 2 = 12 \implies (2 \times 12)$ and $(12 \times 2)$
- $24 \div 3 = 8 \implies (3 \times 8)$ and $(8 \times 3)$
- $24 \div 4 = 6 \implies (4 \times 6)$ and $(6 \times 4)$
| Possible Orders ($m \times n$) |
|---|
| $1 \times 24, 24 \times 1, 2 \times 12, 12 \times 2, 3 \times 8, 8 \times 3, 4 \times 6, 6 \times 4$ |
Step 3: Analysis for 13 Elements
Since 13 is a prime number, its only divisors are 1 and itself [Per the Fundamental Theorem of Arithmetic]. Therefore, the only possible factor pairs are:
- $13 \div 1 = 13 \implies (1 \times 13)$ and $(13 \times 1)$
Step 4: Geometric Representation of Matrix Structure
Final Solution: For 24 elements, the possible orders are: $1 \times 24, 24 \times 1, 2 \times 12, 12 \times 2, 3 \times 8, 8 \times 3, 4 \times 6, 6 \times 4$. For 13 elements, the possible orders are: $1 \times 13$ and $13 \times 1$.
More Questions from Class 12 Mathematics Matrices EXERCISE 3.1
- Q1(i): In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, write: (i) The order of the matrix
- Q1(ii): In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, write: (ii) The number of elements
- Q1(iii): In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, write: (iii) Write the elements $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$.
- Q3: If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
- Q4(i): Construct a $2 \times 2$ matrix, $A = [a_{ij}]$, whose elements are given by: (i) $a_{ij} = \frac{(i+j)^2}{2}$
- Q4(ii): Construct a $2 \times 2$ matrix, $A = [a_{ij}]$, whose elements are given by: (ii) $a_{ij} = \frac{i}{j}$
- Q4(iii): Construct a $2 \times 2$ matrix, $A = [a_{ij}]$, whose elements are given by: (iii) $a_{ij} = \frac{(i+2j)^2}{2}$
- Q5(i): Construct a $3 \times 4$ matrix, whose elements are given by: (i) $a_{ij} = \frac{1}{2}|-3i + j|$
- Q5(ii): Construct a $3 \times 4$ matrix, whose elements are given by: (ii) $a_{ij} = 2i-j$
- Q6(i): Find the values of $x, y$ and $z$ from the following equations: (i) $\begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}$
- Q6(ii): Find the values of $x, y$ and $z$ from the following equations: (ii) $\begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}$
- Q6(iii): Find the values of $x, y$ and $z$ from the following equations: (iii) $\begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}$
- Q7: Find the value of $a, b, c$ and $d$ from the equation: $\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$
- Q8: $A = [a_{ij}]_{m \times n}$ is a square matrix, if
- Q9: Which of the given values of $x$ and $y$ make the following pair of matrices equal $\begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix}$, $\begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix}$
- Q10: The number of all possible matrices of order $3 \times 3$ with each entry 0 or 1 is:
CBSE Solutions for Class 12 Mathematics Matrices
Chapters in CBSE - Class 12 Mathematics
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