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Q3:
If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Solution :
Step 1: Theoretical Foundation
An $m \times n$ matrix is defined by its dimensions, where $m$ represents the number of rows and $n$ represents the number of columns. The total number of elements $N$ in such a matrix is given by the product of the number of rows and the number of columns: $N = m \times n$ [By the Definition of Matrix Order, where $m, n \in \mathbb{N}$]. To find all possible orders for a matrix with $N$ elements, we must identify all possible ordered pairs $(m, n)$ such that their product equals $N$.
Step 2: Analysis for 18 Elements
Given $N = 18$, we seek all pairs of natural numbers $(m, n)$ such that $m \times n = 18$. We identify these by listing the factors of 18:
- $1 \times 18 = 18$
- $2 \times 9 = 18$
- $3 \times 6 = 18$
- $6 \times 3 = 18$
- $9 \times 2 = 18$
- $18 \times 1 = 18$
| Number of Rows ($m$) | Number of Columns ($n$) | Order ($m \times n$) |
|---|---|---|
| 1 | 18 | $1 \times 18$ |
| 2 | 9 | $2 \times 9$ |
| 3 | 6 | $3 \times 6$ |
| 6 | 3 | $6 \times 3$ |
| 9 | 2 | $9 \times 2$ |
| 18 | 1 | $18 \times 1$ |
Step 3: Analysis for 5 Elements
Given $N = 5$, we identify the factors of 5. Since 5 is a prime number [By the Fundamental Theorem of Arithmetic], its only divisors are 1 and 5.
- $1 \times 5 = 5$
- $5 \times 1 = 5$
Step 4: Conclusion
The possible orders are determined by the set of all divisor pairs of the given integer $N$.
Final Solution: For 18 elements, the possible orders are $1 \times 18$, $2 \times 9$, $3 \times 6$, $6 \times 3$, $9 \times 2$, and $18 \times 1$. For 5 elements, the possible orders are $1 \times 5$ and $5 \times 1$.
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 $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$.
- Q2: If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 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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