Let $\mathrm{R}=\left\{\left(\begin{array}{lll}\mathrm{a} & 3 & \mathrm{~b} \\ \mathrm{c} & 2 & \mathrm{~d} \\ 0 & 5 & 0\end{array}\right): \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \in\{0,3,5,7,11,13,17,19\}\right\}$. Then the number of invertible matrices in R is

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Matrices

Let

R = ⎩ ⎨ ⎧ a c 0 325 b d 0 : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19} ⎭ ⎬ ⎫

. Then the number of invertible matrices in R is

Answer: 3780

A matrix in R is invertible iff its determinant is nonzero.

Expanding along the third row:

det = 0 - 5 (a d - b c) + 0 = - 5 (a d - b c)

So the matrix is invertible iff

a d \neq = b c

.

Total matrices:

8^{4} = 4096

**Count when

a d = b c

:**

*Case 1:*

a d = b c = 0

$a d = 0$ : either $a = 0$ or $d = 0$ (or both). Count: $8^{2} - 7^{2} = 15$ (total minus both nonzero)
$b c = 0$ : similarly 15 ways
Total: $15 \times 15 = 225$ ways

*Case 2:*

a d = b c \neq = 0

Sub-case: $a = b = c = d$ (all equal, nonzero): 7 ways
Sub-case: $a \neq = d, b \neq = c$ but $a d = b c$ : Choose 2 distinct values from 7 for ${a, d}$ and 2 for ${b, c}$ , then assign so product equals: $(2 7) \times 2 \times 2 = 21 \times 4 = 84$ ways
Total: $7 + 84 = 91$ ways

Total non-invertible:

225 + 91 = 316

Invertible:

4096 - 316 = 3780

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Integer answer