![62. If [ x y p q u v ] R = [ 0 0 0 0 0 0 0 0 1 ] then the order of R is (A) 2 × 3 (B) 3 × 2 (C) 2 × 2 (D) 3 × 3](https://www.letstalkacademy.com/wp-content/uploads/2026/07/maths-part-2-_page-0007.jpg)
62. If
[
x y
p q
u v
]
R =
[
0 0 0
0 0 0
0 0 1
]
then the order of R is
(A) 2 × 3
(B) 3 × 2
(C) 2 × 2
(D) 3 × 3
How to Find the Order of Matrix R
Although this question appears simple, it actually tests whether a student understands the rules governing matrix multiplication. Many students directly count rows and columns of the visible matrices without applying the multiplication rule properly, which often leads to incorrect answers. Therefore, before solving this question, it is essential to understand how the dimensions of matrices behave during multiplication.
Correct Answer
Option (A): 2 × 3
Understanding the Given Matrices
The first step is to determine the order of the matrices that are already given in the question.
The leftmost matrix is
[
x y
p q
u v
]
This matrix clearly has three rows and two columns.
Therefore, its order is
3 × 2
The matrix on the right-hand side is
[
0 0 0
0 0 0
0 0 1
]
This matrix has
- 3 rows
- 3 columns
Hence, its order is
3 × 3.
Concept Behind Matrix Multiplication
Before finding the order of matrix R, we must recall the most important rule of matrix multiplication.
If
A(m × n) × B(n × p)
then the multiplication is possible only when the number of columns of the first matrix equals the number of rows of the second matrix.
After multiplication, the resulting matrix has the order
m × p.
This is one of the most fundamental rules in Linear Algebra and is tested repeatedly in competitive examinations.
Applying the Matrix Multiplication Rule
The given equation is
(3 × 2) × R = (3 × 3)
Suppose the order of matrix R is
2 × n
The number of columns of the first matrix is 2.
Therefore, the number of rows of matrix R must also be 2.
Next, observe the order of the final product.
The resulting matrix is of order 3 × 3.
Since the result always keeps
- the rows of the first matrix, and
- the columns of the second matrix,
matrix R must contain 3 columns.
Therefore,
Order of R = 2 × 3.
Mathematical Verification
Let
A = 3 × 2
R = 2 × 3
Then
(3 × 2) × (2 × 3)
Since the inner dimensions are equal (2 = 2), multiplication is possible.
The resulting matrix becomes
3 × 3
which exactly matches the matrix given in the question.
Hence, our answer is mathematically verified.
Explanation of Every Option
Option (A): 2 × 3
This option is correct. Matrix multiplication requires the number of rows of matrix R to be equal to the number of columns of the first matrix, which is 2. Since the final matrix has three columns, matrix R must have three columns as well. Therefore, its order becomes 2 × 3.
Option (B): 3 × 2
This option is incorrect because multiplying a matrix of order 3 × 2 with another matrix of order 3 × 2 is not possible. The inner dimensions would be 2 and 3, which are not equal. Since matrix multiplication cannot even be performed, this option is invalid.
Option (C): 2 × 2
This option is also incorrect. Although multiplication is possible because the inner dimensions match (2 = 2), the resulting matrix would have order 3 × 2, not 3 × 3. Therefore, it does not satisfy the given equation.
Option (D): 3 × 3
This option is incorrect because the first matrix has two columns, whereas this matrix has three rows. Since the inner dimensions (2 and 3) are unequal, multiplication cannot be performed. Therefore, this option violates the basic rule of matrix multiplication.
General Formula to Remember
If
A(m × n) × B(n × p)
then
AB = m × p
where
- The inner dimensions (n) must be equal.
- The resulting matrix takes the outer dimensions (m × p).
This simple rule can solve almost every question related to determining the order of matrices.
Additional Example for Better Understanding
Suppose a matrix of order 4 × 5 is multiplied by another matrix to produce a matrix of order 4 × 7. What should be the order of the second matrix?
Using the multiplication rule, the second matrix must have 5 rows so that multiplication is possible. Since the final matrix has 7 columns, the second matrix must also have 7 columns.
Therefore, the required order of the second matrix is 5 × 7.
This example follows exactly the same reasoning used in the given question and reinforces the concept.
Key Takeaways
The order of a matrix is determined by counting its rows and columns. During matrix multiplication, the number of columns of the first matrix must equal the number of rows of the second matrix. The product always retains the number of rows from the first matrix and the number of columns from the second matrix. Applying this rule carefully makes it easy to solve matrix-order questions within a few seconds during competitive examinations.
Final Answer
The first matrix has order 3 × 2, while the resulting matrix has order 3 × 3. Therefore, matrix R must have order 2 × 3 so that the multiplication is valid and produces the required result.
Correct Option: (A) 2 × 3
