JEE Advanced · 2017 · 21 May 2017 · Paper 2 · Matrices

How many

3 \times 3

matrices

M

with entries from

{0, 1, 2}

are there, for which the sum of the diagonal entries of

M^{T} M

is 5?

Answer: (B)

Solution

Let

M = [k_{ij}]

be a

3 \times 3

matrix. The diagonal entries of

M^{T} M

are the sums of squares of entries in each column of

M

.

The sum of diagonal entries of

M^{T} M

equals:

i = 1 \sum 3 j = 1 \sum 3 k_{ij}^{2}

We need

\sum k_{ij}^{2} = 5

where each

k_{ij} \in {0, 1, 2}

.

Since

0^{2} = 0

1^{2} = 1

2^{2} = 4

, the only ways to get a sum of 5 are:

Five entries equal to 1, rest equal to 0: $(5 9)$ ways
One entry equal to 2, one entry equal to 1, rest equal to 0: $(1 9) \times (1 8)$ ways

Total:

(5 9) + 9 \times 8 = 126 + 72 = 198

JEE Advanced · 2017 · 21 May 2017 · Paper 2 · Single correct option