GATE Overflow Test Series | Mock GATE | Test 3 | Question: 36

Question

Consider a matrix Type-$A$ such that $[A_{ij}]_{3 \times 3}=X_{\text{base}5}.$ The elements of matrix are also dependent on their position as, if $(i+j)\%2=0$, then $A_{ij}$ is even and else $A_{ij}$ is odd. Calculate the number of non-symmetric matrices of Type-$A$ ________

Comments

The elements of matrix are also dependent on their position as, if $(i+j)$, then $A_{ij}$ is even...

It’s missing the critical word “even” after $(i+j)$?

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nice question

Answer 1

$X_{\text{base}5}$ means that the elements of matrix $A$ are of base $5$, i.e., the numbers that can be used will be from $\{0,1,2,3,4\}$

$A = \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix}$

Since the element of matrix $P$ is dependent on their position i.e., $(i+j)\%2=0$, then the element at that position is even otherwise odd.

So, for $A_{11}$, $1+1$ is even so even number will be here and allowed even numbers are $0,2,4$.

Hence $A_{11}$ can have $3$ numbers.

Now the number of non-symmetric matrices $=$ total number of matrices possible $-$ number of symmetric matrices

For total number of matrices we have

• $A_{11} = 3 (0,2,4)$
• $A_{12} = 2 (1,3)$
• $A_{13} = 3 (0,2,4)$
• $A_{21} = 2 (1,3)$
• $A_{22} = 3 (0,2,4)$
• $A_{23} = 2 (1,3)$
• $A_{31} = 3 (0,2,4)$
• $A_{32} = 2 (1,3)$
• $A_{33} = 3 (0,2,4)$

So, total number of matrices are $3^5 \times 2^4 = 3888$

For total number of symmetric matrices.

The number, which is present at $A_{21}$ will be same as $A_{12}$

So, for $A_{12} = 1 (A_{21})$

$A_{13} = 1 (A_{31})$

$A_{23} = 1 (A_{32})$

So total number of matrices are $3^4 \times 2^2 = 324$

Therefore total number of non-symmetric matrices are $= 3888-324 = 3564.$

Comments

@wander in base 5 also,we have 0,1,2,3,4 only

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exactly any no can be represented in base 5. why should it be in single base5 digit only

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@Sachin Mittal 1 sir, Don’t you think instead of base 5 it should be mod 5….